author | Kevin Walker <kevin@canyon23.net> |
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%!TEX root = ../blob1.tex |
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\def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip} |
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\def\mmpar#1#2#3{\smallskip\noindent{\bf #1} (#2). {\it #3} \smallskip} |
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\section{\texorpdfstring{$n$}{n}-categories and their modules} |
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\label{sec:ncats} |
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\subsection{Definition of \texorpdfstring{$n$}{n}-categories} |
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\label{ss:n-cat-def} |
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Before proceeding, we need more appropriate definitions of $n$-categories, |
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$A_\infty$ $n$-categories, as well as modules for these, and tensor products of these modules. |
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(As is the case throughout this paper, by ``$n$-category" we mean some notion of |
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a ``weak" $n$-category with ``strong duality".) |
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Compared to other definitions in the literature, |
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the definitions presented below tie the categories more closely to the topology |
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and avoid combinatorial questions about, for example, finding a minimal sufficient |
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collection of generalized associativity axioms; we prefer maximal sets of axioms to minimal sets. |
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It is easy to show that examples of topological origin |
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(e.g.\ categories whose morphisms are maps into spaces or decorated balls, or bordism categories) |
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satisfy our axioms. |
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To show that examples of a more purely algebraic origin satisfy our axioms, |
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one would typically need the combinatorial |
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results that we have avoided here. |
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See \S\ref{n-cat-names} for a discussion of $n$-category terminology. |
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%\nn{Say something explicit about Lurie's work here? |
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%It seems like this was something that Dan Freed wanted explaining when we talked to him in Aspen} |
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\medskip |
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The axioms for an $n$-category are spread throughout this section. |
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Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms}, |
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\ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product}, \ref{axiom:extended-isotopies} and \ref{axiom:splittings}. |
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For an enriched $n$-category we add Axiom \ref{axiom:enriched}. |
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For an $A_\infty$ $n$-category, we replace |
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Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}. |
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Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms |
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for $k{-}1$-morphisms. |
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Readers who prefer things to be presented in a strictly logical order should read this |
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subsection $n{+}1$ times, first setting $k=0$, then $k=1$, and so on until they reach $k=n$. |
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\medskip |
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There are many existing definitions of $n$-categories, with various intended uses. |
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In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. |
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Generally, these sets are indexed by instances of a certain typical shape. |
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Some $n$-category definitions model $k$-morphisms on the standard bihedron (interval, bigon, and so on). |
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Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$, |
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a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$, |
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and so on. |
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(This allows for strict associativity; see \cite{ulrike-tillmann-2008,0909.2212}.) |
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Still other definitions (see, for example, \cite{MR2094071}) |
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model the $k$-morphisms on more complicated combinatorial polyhedra. |
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For our definition, we will allow our $k$-morphisms to have any shape, so long as it is |
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homeomorphic to the standard $k$-ball. |
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Thus we associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic |
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to the standard $k$-ball. |
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By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
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standard $k$-ball. |
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We {\it do not} assume that it is equipped with a |
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preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below. |
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Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on |
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the boundary), we want a corresponding |
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bijection of sets $f:\cC_k(X)\to \cC_k(Y)$. |
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(This will imply ``strong duality", among other things.) Putting these together, we have |
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\begin{axiom}[Morphisms] |
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\label{axiom:morphisms} |
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For each $0 \le k \le n$, we have a functor $\cC_k$ from |
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the category of $k$-balls and |
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homeomorphisms to the category of sets and bijections. |
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\end{axiom} |
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(Note: We often omit the subscript $k$.) |
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We are being deliberately vague about what flavor of $k$-balls |
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we are considering. |
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They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$. |
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They could be PL or smooth. |
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%\nn{need to check whether this makes much difference} |
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(If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need |
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to be fussier about corners and boundaries.) |
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For each flavor of manifold there is a corresponding flavor of $n$-category. |
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For simplicity, we will concentrate on the case of PL unoriented manifolds. |
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(An interesting open question is whether the techniques of this paper can be adapted to topological |
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manifolds and plain, merely continuous homeomorphisms. |
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The main obstacles are proving a version of Lemma \ref{basic_adaptation_lemma} and adapting the |
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transversality arguments used in Lemma \ref{lem:colim-injective}.) |
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An ambitious reader may want to keep in mind two other classes of balls. |
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The first is balls equipped with a map to some other space $Y$ (c.f. \cite{MR2079378}). |
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This will be used below (see the end of \S \ref{ss:product-formula}) to describe the blob complex of a fiber bundle with |
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base space $Y$. |
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The second is balls equipped (partially defined) sections of the tangent bundle, or the frame |
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bundle (i.e.\ framed balls), or more generally some partial flag bundle associated to the tangent bundle. |
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These can be used to define categories with less than the ``strong" duality we assume here, |
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though we will not develop that idea in this paper. |
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Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries |
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of morphisms). |
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The 0-sphere is unusual among spheres in that it is disconnected. |
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Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
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(Actually, this is only true in the oriented case, with 1-morphisms parameterized |
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by {\it oriented} 1-balls.) |
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For $k>1$ and in the presence of strong duality the division into domain and range makes less sense. |
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For example, in a pivotal tensor category, there are natural isomorphisms $\Hom{}{A}{B \tensor C} \isoto \Hom{}{B^* \tensor A}{C}$, etc. |
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(sometimes called ``Frobenius reciprocity''), which canonically identify all the morphism spaces which have the same boundary. |
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We prefer not to make the distinction in the first place. |
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Instead, we will combine the domain and range into a single entity which we call the |
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boundary of a morphism. |
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Morphisms are modeled on balls, so their boundaries are modeled on spheres. |
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In other words, we need to extend the functors $\cC_{k-1}$ from balls to spheres, for |
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$1\le k \le n$. |
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At first it might seem that we need another axiom |
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(more specifically, additional data) for this, but in fact once we have |
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all the axioms in this subsection for $0$ through $k-1$ we can use a colimit |
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construction, as described in \S\ref{ss:ncat-coend} below, to extend $\cC_{k-1}$ |
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to spheres (and any other manifolds): |
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\begin{lem} |
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\label{lem:spheres} |
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For each $1 \le k \le n$, we have a functor $\cl{\cC}_{k-1}$ from |
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the category of $k{-}1$-spheres and |
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homeomorphisms to the category of sets and bijections. |
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\end{lem} |
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We postpone the proof of this result until after we've actually given all the axioms. |
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Note that defining this functor for fixed $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, |
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along with the data described in the other axioms for smaller values of $k$. |
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Of course, Lemma \ref{lem:spheres}, as stated, is satisfied by the trivial functor. |
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What we really mean is that there exists a functor which interacts with the other data of $\cC$ as specified |
143 |
in the axioms below. |
|
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\begin{axiom}[Boundaries]\label{nca-boundary} |
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For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
187 | 148 |
These maps, for various $X$, comprise a natural transformation of functors. |
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\end{axiom} |
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|
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Note that the first ``$\bd$" above is part of the data for the category, |
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while the second is the ordinary boundary of manifolds. |
333 | 153 |
Given $c\in\cl{\cC}(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$. |
94 | 154 |
|
155 |
\medskip |
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In order to simplify the exposition we have concentrated on the case of |
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unoriented PL manifolds and avoided the question of what exactly we mean by |
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the boundary of a manifold with extra structure, such as an oriented manifold. |
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In general, all manifolds of dimension less than $n$ should be equipped with the germ |
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of a thickening to dimension $n$, and this germ should carry whatever structure we have |
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on $n$-manifolds. |
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In addition, lower dimensional manifolds should be equipped with a framing |
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of their normal bundle in the thickening; the framing keeps track of which |
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side (iterated) bounded manifolds lie on. |
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For example, the boundary of an oriented $n$-ball |
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should be an $n{-}1$-sphere equipped with an orientation of its once stabilized tangent |
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bundle and a choice of direction in this bundle indicating |
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which side the $n$-ball lies on. |
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|
94 | 171 |
\medskip |
172 |
||
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We have just argued that the boundary of a morphism has no preferred splitting into |
|
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domain and range, but the converse meets with our approval. |
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That is, given compatible domain and range, we should be able to combine them into |
|
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the full boundary of a morphism. |
402 | 177 |
The following lemma will follow from the colimit construction used to define $\cl{\cC}_{k-1}$ |
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on spheres. |
94 | 179 |
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\begin{lem}[Boundary from domain and range] |
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\label{lem:domain-and-range} |
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Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$, |
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$B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}). |
333 | 184 |
Let $\cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2)$ denote the fibered product of the |
185 |
two maps $\bd: \cC(B_i)\to \cl{\cC}(E)$. |
|
187 | 186 |
Then we have an injective map |
94 | 187 |
\[ |
402 | 188 |
\gl_E : \cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2) \into \cl{\cC}(S) |
94 | 189 |
\] |
187 | 190 |
which is natural with respect to the actions of homeomorphisms. |
333 | 191 |
(When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product |
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becomes a normal product.) |
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\end{lem} |
94 | 194 |
|
774 | 195 |
\begin{figure}[t] \centering |
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\begin{tikzpicture}[%every label/.style={green} |
333 | 197 |
] |
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\node[fill=black, circle, label=below:$E$, inner sep=1.5pt](S) at (0,0) {}; |
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\node[fill=black, circle, label=above:$E$, inner sep=1.5pt](N) at (0,2) {}; |
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\draw (S) arc (-90:90:1); |
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\draw (N) arc (90:270:1); |
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\node[left] at (-1,1) {$B_1$}; |
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\node[right] at (1,1) {$B_2$}; |
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\end{tikzpicture} |
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\caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure} |
179 | 206 |
|
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Note that we insist on injectivity above. |
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The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}. |
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%\nn{we might want a more official looking proof...} |
109 | 210 |
|
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We do not insist on surjectivity of the gluing map, since this is not satisfied by all of the examples |
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we are trying to axiomatize. |
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If our $k$-morphisms $\cC(X)$ are labeled cell complexes embedded in $X$ (c.f. Example \ref{ex:traditional-n-categories} below), then a $k$-morphism is |
837 | 214 |
in the image of the gluing map precisely when the cell complex is in general position |
215 |
with respect to $E$. On the other hand, in categories based on maps to a target space (c.f. Example \ref{ex:maps-to-a-space} below) the gluing map is always surjective. |
|
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216 |
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If $S$ is a 0-sphere (the case $k=1$ above), then $S$ can be identified with the {\it disjoint} union |
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of two 0-balls $B_1$ and $B_2$ and the colimit construction $\cl{\cC}(S)$ can be identified |
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with the (ordinary, not fibered) product $\cC(B_1) \times \cC(B_2)$. |
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Let $\cl{\cC}(S)\trans E$ denote the image of $\gl_E$. |
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We will refer to elements of $\cl{\cC}(S)\trans E$ as ``splittable along $E$" or ``transverse to $E$". |
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When the gluing map is surjective every such element is splittable. |
109 | 224 |
|
195 | 225 |
If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ |
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as above, then we define $\cC(X)\trans E = \bd^{-1}(\cl{\cC}(\bd X)\trans E)$. |
195 | 227 |
|
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We will call the projection $\cl{\cC}(S)\trans E \to \cC(B_i)$ given by the composition |
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$$\cl{\cC}(S)\trans E \xrightarrow{\gl^{-1}} \cC(B_1) \times \cC(B_2) \xrightarrow{\pr_i} \cC(B_i)$$ |
110 | 230 |
a {\it restriction} map and write $\res_{B_i}(a)$ |
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(or simply $\res(a)$ when there is no ambiguity), for $a\in \cl{\cC}(S)\trans E$. |
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More generally, we also include under the rubric ``restriction map" |
195 | 233 |
the boundary maps of Axiom \ref{nca-boundary} above, |
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234 |
another class of maps introduced after Axiom \ref{nca-assoc} below, as well as any composition |
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235 |
of restriction maps. |
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In particular, we have restriction maps $\cC(X)\trans E \to \cC(B_i)$ |
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defined as the composition of the boundary with the first restriction map described above: |
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238 |
$$ |
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239 |
\cC(X) \trans E \xrightarrow{\bdy} \cl{\cC}(\bdy X)\trans E \xrightarrow{\res} \cC(B_i) |
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240 |
.$$ |
195 | 241 |
These restriction maps can be thought of as |
242 |
domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$. |
|
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These restriction maps in fact have their image in the subset $\cC(B_i)\trans E$, |
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and so to emphasize this we will sometimes write the restriction map as $\cC(X)\trans E \to \cC(B_i)\trans E$. |
94 | 245 |
|
246 |
||
247 |
Next we consider composition of morphisms. |
|
248 |
For $n$-categories which lack strong duality, one usually considers |
|
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$k$ different types of composition of $k$-morphisms, each associated to a different ``direction". |
94 | 250 |
(For example, vertical and horizontal composition of 2-morphisms.) |
251 |
In the presence of strong duality, these $k$ distinct compositions are subsumed into |
|
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one general type of composition which can be in any direction. |
94 | 253 |
|
187 | 254 |
\begin{axiom}[Composition] |
560 | 255 |
\label{axiom:composition} |
187 | 256 |
Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
179 | 257 |
and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
103 | 258 |
Let $E = \bd Y$, which is a $k{-}2$-sphere. |
94 | 259 |
Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
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We have restriction (domain or range) maps $\cC(B_i)\trans E \to \cC(Y)$. |
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Let $\cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E$ denote the fibered product of these two maps. |
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We have a map |
94 | 263 |
\[ |
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\gl_Y : \cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E \to \cC(B)\trans E |
94 | 265 |
\] |
266 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
267 |
to the intersection of the boundaries of $B$ and $B_i$. |
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If $k < n$ |
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we require that $\gl_Y$ is injective. |
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%(For $k=n$ see below.) |
187 | 271 |
\end{axiom} |
94 | 272 |
|
774 | 273 |
\begin{figure}[t] \centering |
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\begin{tikzpicture}[%every label/.style={green}, |
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x=1.5cm,y=1.5cm] |
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\node[fill=black, circle, label=below:$E$, inner sep=2pt](S) at (0,0) {}; |
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\node[fill=black, circle, label=above:$E$, inner sep=2pt](N) at (0,2) {}; |
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\draw (S) arc (-90:90:1); |
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\draw (N) arc (90:270:1); |
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\draw (N) -- (S); |
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\node[left] at (-1/4,1) {$B_1$}; |
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\node[right] at (1/4,1) {$B_2$}; |
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\node at (1/6,3/2) {$Y$}; |
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284 |
\end{tikzpicture} |
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285 |
\caption{From two balls to one ball.}\label{blah5}\end{figure} |
179 | 286 |
|
195 | 287 |
\begin{axiom}[Strict associativity] \label{nca-assoc} |
187 | 288 |
The composition (gluing) maps above are strictly associative. |
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289 |
Given any splitting of a ball $B$ into smaller balls |
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$$\bigsqcup B_i \to B,$$ |
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291 |
any sequence of gluings (in the sense of Definition \ref{defn:gluing-decomposition}, where all the intermediate steps are also disjoint unions of balls) yields the same result. |
187 | 292 |
\end{axiom} |
102 | 293 |
|
774 | 294 |
\begin{figure}[t] |
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$$\mathfig{.65}{ncat/strict-associativity}$$ |
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296 |
\caption{An example of strict associativity.}\label{blah6}\end{figure} |
179 | 297 |
|
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298 |
We'll use the notation $a\bullet b$ for the glued together field $\gl_Y(a, b)$. |
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299 |
In the other direction, we will call the projection from $\cC(B)\trans E$ to $\cC(B_i)\trans E$ |
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300 |
a restriction map (one of many types of map so called) and write $\res_{B_i}(a)$ for $a\in \cC(B)\trans E$. |
195 | 301 |
%Compositions of boundary and restriction maps will also be called restriction maps. |
302 |
%For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a |
|
303 |
%restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$. |
|
110 | 304 |
|
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305 |
We will write $\cC(B)\trans Y$ for the image of $\gl_Y$ in $\cC(B)$. |
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306 |
We will call elements of $\cC(B)\trans Y$ morphisms which are |
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307 |
``splittable along $Y$'' or ``transverse to $Y$''. |
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308 |
We have $\cC(B)\trans Y \sub \cC(B)\trans E \sub \cC(B)$. |
109 | 309 |
|
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310 |
More generally, let $\alpha$ be a splitting of $X$ into smaller balls. |
193 | 311 |
Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from |
312 |
the smaller balls to $X$. |
|
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We say that elements of $\cC(X)_\alpha$ are morphisms which are ``splittable along $\alpha$". |
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314 |
In situations where the splitting is notationally anonymous, we will write |
193 | 315 |
$\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to) |
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316 |
the unnamed splitting. |
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317 |
If $\beta$ is a ball decomposition of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cl{\cC}(\bd X)_\beta)$; |
193 | 318 |
this can also be denoted $\cC(X)\spl$ if the context contains an anonymous |
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319 |
decomposition of $\bd X$ and no competing splitting of $X$. |
192 | 320 |
|
321 |
The above two composition axioms are equivalent to the following one, |
|
102 | 322 |
which we state in slightly vague form. |
323 |
||
324 |
\xxpar{Multi-composition:} |
|
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325 |
{Given any splitting $B_1 \sqcup \cdots \sqcup B_m \to B$ of a $k$-ball |
102 | 326 |
into small $k$-balls, there is a |
327 |
map from an appropriate subset (like a fibered product) |
|
193 | 328 |
of $\cC(B_1)\spl\times\cdots\times\cC(B_m)\spl$ to $\cC(B)\spl$, |
95 | 329 |
and these various $m$-fold composition maps satisfy an |
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330 |
operad-type strict associativity condition (Figure \ref{fig:operad-composition}).} |
179 | 331 |
|
774 | 332 |
\begin{figure}[t] |
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333 |
$$\mathfig{.8}{ncat/operad-composition}$$ |
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334 |
\caption{Operad composition and associativity}\label{fig:operad-composition}\end{figure} |
95 | 335 |
|
336 |
The next axiom is related to identity morphisms, though that might not be immediately obvious. |
|
337 |
||
343 | 338 |
\begin{axiom}[Product (identity) morphisms, preliminary version] |
339 |
For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$, |
|
340 |
usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. |
|
341 |
These maps must satisfy the following conditions. |
|
342 |
\begin{enumerate} |
|
343 |
\item |
|
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344 |
If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are homeomorphisms such that the diagram |
343 | 345 |
\[ \xymatrix{ |
346 |
X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\ |
|
347 |
X \ar[r]^{f} & X' |
|
348 |
} \] |
|
349 |
commutes, then we have |
|
350 |
\[ |
|
351 |
\tilde{f}(a\times D) = f(a)\times D' . |
|
352 |
\] |
|
353 |
\item |
|
354 |
Product morphisms are compatible with gluing (composition) in both factors: |
|
355 |
\[ |
|
356 |
(a'\times D)\bullet(a''\times D) = (a'\bullet a'')\times D |
|
357 |
\] |
|
358 |
and |
|
359 |
\[ |
|
360 |
(a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') . |
|
361 |
\] |
|
362 |
\item |
|
363 |
Product morphisms are associative: |
|
364 |
\[ |
|
365 |
(a\times D)\times D' = a\times (D\times D') . |
|
366 |
\] |
|
367 |
(Here we are implicitly using functoriality and the obvious homeomorphism |
|
368 |
$(X\times D)\times D' \to X\times(D\times D')$.) |
|
369 |
\item |
|
370 |
Product morphisms are compatible with restriction: |
|
371 |
\[ |
|
372 |
\res_{X\times E}(a\times D) = a\times E |
|
373 |
\] |
|
374 |
for $E\sub \bd D$ and $a\in \cC(X)$. |
|
375 |
\end{enumerate} |
|
376 |
\end{axiom} |
|
377 |
||
378 |
We will need to strengthen the above preliminary version of the axiom to allow |
|
379 |
for products which are ``pinched" in various ways along their boundary. |
|
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380 |
(See Figure \ref{pinched_prods}.) |
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381 |
\begin{figure}[t] |
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382 |
$$ |
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383 |
\begin{tikzpicture}[baseline=0] |
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384 |
\begin{scope} |
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385 |
\path[clip] (0,0) arc (135:45:4) arc (-45:-135:4); |
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386 |
\draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4); |
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387 |
\foreach \x in {0, 0.5, ..., 6} { |
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388 |
\draw[green!50!brown] (\x,-2) -- (\x,2); |
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389 |
} |
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390 |
\end{scope} |
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391 |
\draw[blue,line width=1.5pt] (0,-3) -- (5.66,-3); |
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392 |
\draw[->,red,line width=2pt] (2.83,-1.5) -- (2.83,-2.5); |
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393 |
\end{tikzpicture} |
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394 |
\qquad \qquad |
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395 |
\begin{tikzpicture}[baseline=-0.15cm] |
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396 |
\begin{scope} |
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397 |
\path[clip] (0,1) arc (90:135:8 and 4) arc (-135:-90:8 and 4) -- cycle; |
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398 |
\draw[blue,line width=2pt] (0,1) arc (90:135:8 and 4) arc (-135:-90:8 and 4) -- cycle; |
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399 |
\foreach \x in {-6, -5.5, ..., 0} { |
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400 |
\draw[green!50!brown] (\x,-2) -- (\x,2); |
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401 |
} |
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402 |
\end{scope} |
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403 |
\draw[blue,line width=1.5pt] (-5.66,-3.15) -- (0,-3.15); |
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404 |
\draw[->,red,line width=2pt] (-2.83,-1.5) -- (-2.83,-2.5); |
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405 |
\end{tikzpicture} |
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406 |
$$ |
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|
407 |
\caption{Examples of pinched products}\label{pinched_prods} |
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|
408 |
\end{figure} |
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409 |
The need for a strengthened version will become apparent in Appendix \ref{sec:comparing-defs} |
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|
410 |
where we construct a traditional 2-category from a disk-like 2-category. |
754
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|
411 |
For example, ``half-pinched" products of 1-balls are used to construct weak identities for 1-morphisms |
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|
412 |
in 2-categories. |
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413 |
We also need fully-pinched products to define collar maps below (see Figure \ref{glue-collar}). |
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414 |
|
343 | 415 |
Define a {\it pinched product} to be a map |
416 |
\[ |
|
417 |
\pi: E\to X |
|
418 |
\] |
|
344 | 419 |
such that $E$ is a $k{+}m$-ball, $X$ is a $k$-ball ($m\ge 1$), and $\pi$ is locally modeled |
343 | 420 |
on a standard iterated degeneracy map |
421 |
\[ |
|
344 | 422 |
d: \Delta^{k+m}\to\Delta^k . |
343 | 423 |
\] |
424 |
(We thank Kevin Costello for suggesting this approach.) |
|
425 |
||
344 | 426 |
Note that for each interior point $x\in X$, $\pi\inv(x)$ is an $m$-ball, |
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427 |
and for each boundary point $x\in\bd X$, $\pi\inv(x)$ is a ball of dimension |
344 | 428 |
$l \le m$, with $l$ depending on $x$. |
343 | 429 |
It is easy to see that a composition of pinched products is again a pinched product. |
430 |
A {\it sub pinched product} is a sub-$m$-ball $E'\sub E$ such that the restriction |
|
431 |
$\pi:E'\to \pi(E')$ is again a pinched product. |
|
432 |
A {union} of pinched products is a decomposition $E = \cup_i E_i$ |
|
433 |
such that each $E_i\sub E$ is a sub pinched product. |
|
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434 |
(See Figure \ref{pinched_prod_unions}.) |
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|
435 |
\begin{figure}[t] |
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|
436 |
$$ |
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437 |
\begin{tikzpicture}[baseline=0] |
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438 |
\begin{scope} |
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439 |
\path[clip] (0,0) arc (135:45:4) arc (-45:-135:4); |
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\draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4); |
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\draw[blue] (0,0) -- (5.66,0); |
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442 |
\foreach \x in {0, 0.5, ..., 6} { |
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\draw[green!50!brown] (\x,-2) -- (\x,2); |
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444 |
} |
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\end{scope} |
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\end{tikzpicture} |
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447 |
\qquad |
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448 |
\begin{tikzpicture}[baseline=0] |
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\begin{scope} |
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\path[clip] (0,-1) rectangle (4,1); |
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\draw[blue,line width=2pt] (0,-1) rectangle (4,1); |
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|
452 |
\draw[blue] (0,0) -- (5,0); |
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|
453 |
\foreach \x in {0, 0.5, ..., 6} { |
93d636f420c7
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changeset
|
454 |
\draw[green!50!brown] (\x,-2) -- (\x,2); |
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|
455 |
} |
93d636f420c7
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|
456 |
\end{scope} |
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|
457 |
\end{tikzpicture} |
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|
458 |
\qquad |
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changeset
|
459 |
\begin{tikzpicture}[baseline=0] |
93d636f420c7
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changeset
|
460 |
\begin{scope} |
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|
461 |
\path[clip] (0,0) arc (135:45:4) arc (-45:-135:4); |
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|
462 |
\draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4); |
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|
463 |
\draw[blue] (2.83,3) circle (3); |
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|
464 |
\foreach \x in {0, 0.5, ..., 6} { |
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|
465 |
\draw[green!50!brown] (\x,-2) -- (\x,2); |
93d636f420c7
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|
466 |
} |
93d636f420c7
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changeset
|
467 |
\end{scope} |
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|
468 |
\end{tikzpicture} |
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changeset
|
469 |
$$ |
93d636f420c7
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changeset
|
470 |
$$ |
93d636f420c7
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changeset
|
471 |
\begin{tikzpicture}[baseline=0] |
93d636f420c7
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changeset
|
472 |
\begin{scope} |
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changeset
|
473 |
\path[clip] (0,-1) rectangle (4,1); |
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|
474 |
\draw[blue,line width=2pt] (0,-1) rectangle (4,1); |
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|
475 |
\draw[blue] (0,-1) -- (4,1); |
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|
476 |
\foreach \x in {0, 0.5, ..., 6} { |
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|
477 |
\draw[green!50!brown] (\x,-2) -- (\x,2); |
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|
478 |
} |
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|
479 |
\end{scope} |
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|
480 |
\end{tikzpicture} |
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|
481 |
\qquad |
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|
482 |
\begin{tikzpicture}[baseline=0] |
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|
483 |
\begin{scope} |
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|
484 |
\path[clip] (0,-1) rectangle (5,1); |
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|
485 |
\draw[blue,line width=2pt] (0,-1) rectangle (5,1); |
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|
486 |
\draw[blue] (1,-1) .. controls (2,-1) and (3,1) .. (4,1); |
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|
487 |
\foreach \x in {0, 0.5, ..., 6} { |
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|
488 |
\draw[green!50!brown] (\x,-2) -- (\x,2); |
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|
489 |
} |
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|
490 |
\end{scope} |
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|
491 |
\end{tikzpicture} |
751
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|
492 |
\qquad |
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changeset
|
493 |
\begin{tikzpicture}[baseline=0] |
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|
494 |
\begin{scope} |
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|
495 |
\path[clip] (0,0) arc (135:45:4) arc (-45:-135:4); |
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|
496 |
\draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4); |
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|
497 |
\draw[blue] (2.82,-5) -- (2.83,5); |
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|
498 |
\foreach \x in {0, 0.5, ..., 6} { |
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|
499 |
\draw[green!50!brown] (\x,-2) -- (\x,2); |
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|
500 |
} |
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|
501 |
\end{scope} |
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|
502 |
\end{tikzpicture} |
364
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503 |
$$ |
808 | 504 |
\caption{Six examples of unions of pinched products}\label{pinched_prod_unions} |
352
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changeset
|
505 |
\end{figure} |
343 | 506 |
|
802
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diff
changeset
|
507 |
Note that $\bd X$ has a (possibly trivial) subdivision according to |
e3ddb8605e32
adding transversality requirement to product morphism axiom
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diff
changeset
|
508 |
the dimension of $\pi\inv(x)$, $x\in \bd X$. |
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|
509 |
Let $\cC(X)\trans{}$ denote the morphisms which are splittable along this subdivision. |
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changeset
|
510 |
|
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adding transversality requirement to product morphism axiom
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diff
changeset
|
511 |
The product axiom will give a map $\pi^*:\cC(X)\trans{}\to \cC(E)$ for each pinched product |
343 | 512 |
$\pi:E\to X$. |
344 | 513 |
Morphisms in the image of $\pi^*$ will be called product morphisms. |
343 | 514 |
Before stating the axiom, we illustrate it in our two motivating examples of $n$-categories. |
515 |
In the case where $\cC(X) = \{f: X\to T\}$, we define $\pi^*(f) = f\circ\pi$. |
|
344 | 516 |
In the case where $\cC(X)$ is the set of all labeled embedded cell complexes $K$ in $X$, |
517 |
define $\pi^*(K) = \pi\inv(K)$, with each codimension $i$ cell $\pi\inv(c)$ labeled by the |
|
518 |
same (traditional) $i$-morphism as the corresponding codimension $i$ cell $c$. |
|
343 | 519 |
|
520 |
||
551
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|
521 |
%\addtocounter{axiom}{-1} |
187 | 522 |
\begin{axiom}[Product (identity) morphisms] |
560 | 523 |
\label{axiom:product} |
344 | 524 |
For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
802
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changeset
|
525 |
there is a map $\pi^*:\cC(X)\trans{}\to \cC(E)$. |
340
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|
526 |
These maps must satisfy the following conditions. |
191
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|
527 |
\begin{enumerate} |
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|
528 |
\item |
344 | 529 |
If $\pi:E\to X$ and $\pi':E'\to X'$ are pinched products, and |
530 |
if $f:X\to X'$ and $\tilde{f}:E \to E'$ are maps such that the diagram |
|
95 | 531 |
\[ \xymatrix{ |
344 | 532 |
E \ar[r]^{\tilde{f}} \ar[d]_{\pi} & E' \ar[d]^{\pi'} \\ |
95 | 533 |
X \ar[r]^{f} & X' |
534 |
} \] |
|
109 | 535 |
commutes, then we have |
536 |
\[ |
|
344 | 537 |
\pi'^*\circ f = \tilde{f}\circ \pi^*. |
109 | 538 |
\] |
191
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diff
changeset
|
539 |
\item |
344 | 540 |
Product morphisms are compatible with gluing (composition). |
541 |
Let $\pi:E\to X$, $\pi_1:E_1\to X_1$, and $\pi_2:E_2\to X_2$ |
|
542 |
be pinched products with $E = E_1\cup E_2$. |
|
752
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parents:
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diff
changeset
|
543 |
(See Figure \ref{pinched_prod_unions}.) |
84bf15233e08
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changeset
|
544 |
Note that $X_1$ and $X_2$ can be identified with subsets of $X$, |
84bf15233e08
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Kevin Walker <kevin@canyon23.net>
parents:
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diff
changeset
|
545 |
but $X_1 \cap X_2$ might not be codimension 1, and indeed we might have $X_1 = X_2 = X$. |
84bf15233e08
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parents:
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diff
changeset
|
546 |
We assume that there is a decomposition of $X$ into balls which is compatible with |
84bf15233e08
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parents:
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diff
changeset
|
547 |
$X_1$ and $X_2$. |
802
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adding transversality requirement to product morphism axiom
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parents:
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diff
changeset
|
548 |
Let $a\in \cC(X)\trans{}$, and let $a_i$ denote the restriction of $a$ to $X_i\sub X$. |
753
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parents:
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diff
changeset
|
549 |
(We assume that $a$ is splittable with respect to the above decomposition of $X$ into balls.) |
344 | 550 |
Then |
109 | 551 |
\[ |
344 | 552 |
\pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) . |
109 | 553 |
\] |
191
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diff
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|
554 |
\item |
344 | 555 |
Product morphisms are associative. |
423 | 556 |
If $\pi:E\to X$ and $\rho:D\to E$ are pinched products then |
109 | 557 |
\[ |
344 | 558 |
\rho^*\circ\pi^* = (\pi\circ\rho)^* . |
109 | 559 |
\] |
191
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diff
changeset
|
560 |
\item |
344 | 561 |
Product morphisms are compatible with restriction. |
562 |
If we have a commutative diagram |
|
563 |
\[ \xymatrix{ |
|
564 |
D \ar@{^(->}[r] \ar[d]_{\rho} & E \ar[d]^{\pi} \\ |
|
565 |
Y \ar@{^(->}[r] & X |
|
566 |
} \] |
|
567 |
such that $\rho$ and $\pi$ are pinched products, then |
|
110 | 568 |
\[ |
344 | 569 |
\res_D\circ\pi^* = \rho^*\circ\res_Y . |
110 | 570 |
\] |
191
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changeset
|
571 |
\end{enumerate} |
187 | 572 |
\end{axiom} |
95 | 573 |
|
343 | 574 |
|
575 |
\medskip |
|
128 | 576 |
|
788
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changeset
|
577 |
|
6a1b6c2de201
more reorganization of n-cat defs
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diff
changeset
|
578 |
|
95 | 579 |
|
788
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parents:
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diff
changeset
|
580 |
%All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |
6a1b6c2de201
more reorganization of n-cat defs
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parents:
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diff
changeset
|
581 |
%The last axiom (below), concerning actions of |
6a1b6c2de201
more reorganization of n-cat defs
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parents:
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diff
changeset
|
582 |
%homeomorphisms in the top dimension $n$, distinguishes the two cases. |
6a1b6c2de201
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parents:
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diff
changeset
|
583 |
|
6a1b6c2de201
more reorganization of n-cat defs
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parents:
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diff
changeset
|
584 |
%We start with the ordinary $n$-category case. |
6a1b6c2de201
more reorganization of n-cat defs
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parents:
787
diff
changeset
|
585 |
|
6a1b6c2de201
more reorganization of n-cat defs
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parents:
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diff
changeset
|
586 |
The next axiom says, roughly, that we have strict associativity in dimension $n$, |
800
d0b9238aad5d
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Kevin Walker <kevin@canyon23.net>
parents:
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diff
changeset
|
587 |
even when we reparametrize our $n$-balls. |
95 | 588 |
|
420 | 589 |
\begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$] |
897
9ba67422f1b9
minor fixes, some typos, some cross-references
Scott Morrison <scott@tqft.net>
parents:
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diff
changeset
|
590 |
\label{axiom:isotopy-preliminary} |
788
6a1b6c2de201
more reorganization of n-cat defs
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parents:
787
diff
changeset
|
591 |
Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which |
6a1b6c2de201
more reorganization of n-cat defs
Kevin Walker <kevin@canyon23.net>
parents:
787
diff
changeset
|
592 |
acts trivially on the restriction $\bd b$ of $b$ to $\bd X$. |
6a1b6c2de201
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parents:
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diff
changeset
|
593 |
(Keep in mind the important special case where $f$ restricted to $\bd X$ is the identity.) |
833 | 594 |
Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which act |
788
6a1b6c2de201
more reorganization of n-cat defs
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parents:
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diff
changeset
|
595 |
trivially on $\bd b$. |
6a1b6c2de201
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parents:
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diff
changeset
|
596 |
Then $f(b) = b$. |
6a1b6c2de201
more reorganization of n-cat defs
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parents:
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diff
changeset
|
597 |
In particular, homeomorphisms which are isotopic to the identity rel boundary act trivially on |
6a1b6c2de201
more reorganization of n-cat defs
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parents:
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diff
changeset
|
598 |
all of $\cC(X)$. |
267 | 599 |
\end{axiom} |
96 | 600 |
|
174 | 601 |
This axiom needs to be strengthened to force product morphisms to act as the identity. |
103 | 602 |
Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. |
96 | 603 |
Let $J$ be a 1-ball (interval). |
721
3ae1a110873b
add definition of collaring homeo, etc.
Kevin Walker <kevin@canyon23.net>
parents:
719
diff
changeset
|
604 |
Let $s_{Y,J}: X\cup_Y (Y\times J) \to X$ be a collaring homeomorphism |
3ae1a110873b
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605 |
(see the end of \S\ref{ss:syst-o-fields}). |
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|
606 |
Here we use $Y\times J$ with boundary entirely pinched. |
96 | 607 |
We define a map |
608 |
\begin{eqnarray*} |
|
609 |
\psi_{Y,J}: \cC(X) &\to& \cC(X) \\ |
|
797
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|
610 |
a & \mapsto & s_{Y,J}(a \bullet ((a|_Y)\times J)) . |
96 | 611 |
\end{eqnarray*} |
142 | 612 |
(See Figure \ref{glue-collar}.) |
774 | 613 |
\begin{figure}[t] |
189 | 614 |
\begin{equation*} |
190
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615 |
\begin{tikzpicture} |
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616 |
\def\rad{1} |
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617 |
\def\srad{0.75} |
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\def\gap{4.5} |
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\foreach \i in {0, 1, 2} { |
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\node(\i) at ($\i*(\gap,0)$) [draw, circle through = {($\i*(\gap,0)+(\rad,0)$)}] {}; |
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\node(\i-small) at (\i.east) [circle through={($(\i.east)+(\srad,0)$)}] {}; |
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\foreach \n in {1,2} { |
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\fill (intersection \n of \i-small and \i) node(\i-intersection-\n) {} circle (2pt); |
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} |
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} |
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\begin{scope}[decoration={brace,amplitude=10,aspect=0.5}] |
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\draw[decorate] (0-intersection-1.east) -- (0-intersection-2.east); |
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\end{scope} |
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630 |
\node[right=1mm] at (0.east) {$a$}; |
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\draw[->] ($(0.east)+(0.75,0)$) -- ($(1.west)+(-0.2,0)$); |
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|
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633 |
\draw (1-small) circle (\srad); |
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\foreach \theta in {90, 72, ..., -90} { |
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\draw[blue] (1) -- ($(1)+(\rad,0)+(\theta:\srad)$); |
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} |
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637 |
\filldraw[fill=white] (1) circle (\rad); |
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\foreach \n in {1,2} { |
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639 |
\fill (intersection \n of 1-small and 1) circle (2pt); |
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640 |
} |
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641 |
\node[below] at (1-small.south) {$a \times J$}; |
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\draw[->] ($(1.east)+(1,0)$) -- ($(2.west)+(-0.2,0)$); |
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|
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\begin{scope} |
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645 |
\path[clip] (2) circle (\rad); |
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\draw[clip] (2.east) circle (\srad); |
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\foreach \y in {1, 0.86, ..., -1} { |
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648 |
\draw[blue] ($(2)+(-1,\y) $)-- ($(2)+(1,\y)$); |
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} |
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650 |
\end{scope} |
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|
651 |
\end{tikzpicture} |
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652 |
\end{equation*} |
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|
653 |
\begin{equation*} |
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654 |
\xymatrix@C+2cm{\cC(X) \ar[r]^(0.45){\text{glue}} & \cC(X \cup \text{collar}) \ar[r]^(0.55){\text{homeo}} & \cC(X)} |
189 | 655 |
\end{equation*} |
656 |
||
657 |
\caption{Extended homeomorphism.}\label{glue-collar}\end{figure} |
|
415
8dedd2914d10
starting to revise ncat section
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parents:
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diff
changeset
|
658 |
We call a map of this form a {\it collar map}. |
96 | 659 |
It can be thought of as the action of the inverse of |
415
8dedd2914d10
starting to revise ncat section
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parents:
411
diff
changeset
|
660 |
a map which projects a collar neighborhood of $Y$ onto $Y$, |
8dedd2914d10
starting to revise ncat section
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parents:
411
diff
changeset
|
661 |
or as the limit of homeomorphisms $X\to X$ which expand a very thin collar of $Y$ |
8dedd2914d10
starting to revise ncat section
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parents:
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diff
changeset
|
662 |
to a larger collar. |
8dedd2914d10
starting to revise ncat section
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parents:
411
diff
changeset
|
663 |
We call the equivalence relation generated by collar maps and homeomorphisms |
8dedd2914d10
starting to revise ncat section
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parents:
411
diff
changeset
|
664 |
isotopic (rel boundary) to the identity {\it extended isotopy}. |
96 | 665 |
|
666 |
The revised axiom is |
|
667 |
||
833 | 668 |
\begin{axiom}[Extended isotopy invariance in dimension $n$] |
187 | 669 |
\label{axiom:extended-isotopies} |
788
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diff
changeset
|
670 |
Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which |
6a1b6c2de201
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diff
changeset
|
671 |
acts trivially on the restriction $\bd b$ of $b$ to $\bd X$. |
6a1b6c2de201
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parents:
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diff
changeset
|
672 |
Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which |
797
40729de8e067
finish fam-o-homeo axiom revisions and discussion
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parents:
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diff
changeset
|
673 |
act trivially on $\bd b$. |
788
6a1b6c2de201
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diff
changeset
|
674 |
Then $f(b) = b$. |
415
8dedd2914d10
starting to revise ncat section
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parents:
411
diff
changeset
|
675 |
In addition, collar maps act trivially on $\cC(X)$. |
266
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diff
changeset
|
676 |
\end{axiom} |
96 | 677 |
|
788
6a1b6c2de201
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diff
changeset
|
678 |
\medskip |
97 | 679 |
|
896
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Initial version of the new splitting axiom.
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parents:
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diff
changeset
|
680 |
We need one additional axiom. |
deeff619087e
Initial version of the new splitting axiom.
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parents:
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diff
changeset
|
681 |
It says, roughly, that given a $k$-ball $X$, $k<n$, and $c\in \cC(X)$, there exist sufficiently many splittings of $c$. |
913
75c1e11d0f25
add remarks about the missing TOP case; searched for all occurrances of "topological" and "continuous" to make sure all other mentions of TOP have been expunged; other minor changes
Kevin Walker <kevin@canyon23.net>
parents:
904
diff
changeset
|
682 |
We use this axiom in the proofs of \ref{lem:d-a-acyclic} and \ref{lem:colim-injective}. |
914
db365e67adf6
finished with plitting axiom stuff (except that now the module definition needs to be updated)
Kevin Walker <kevin@canyon23.net>
parents:
913
diff
changeset
|
683 |
The analogous axiom for systems of fields is used in the proof of \ref{small-blobs-b}. |
896
deeff619087e
Initial version of the new splitting axiom.
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parents:
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diff
changeset
|
684 |
All of the examples of (disk-like) $n$-categories we consider in this paper satisfy the axiom, but |
deeff619087e
Initial version of the new splitting axiom.
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parents:
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diff
changeset
|
685 |
nevertheless we feel that it is too strong. |
deeff619087e
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diff
changeset
|
686 |
In the future we would like to see this provisional version of the axiom replaced by something less restrictive. |
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Initial version of the new splitting axiom.
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parents:
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diff
changeset
|
687 |
|
897
9ba67422f1b9
minor fixes, some typos, some cross-references
Scott Morrison <scott@tqft.net>
parents:
896
diff
changeset
|
688 |
We give two alternate versions of the axiom, one better suited for smooth examples, and one better suited to PL examples. |
896
deeff619087e
Initial version of the new splitting axiom.
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changeset
|
689 |
|
deeff619087e
Initial version of the new splitting axiom.
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diff
changeset
|
690 |
\begin{axiom}[Splittings] |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
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diff
changeset
|
691 |
\label{axiom:splittings} |
deeff619087e
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parents:
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diff
changeset
|
692 |
Let $c\in \cC_k(X)$, with $0\le k < n$. |
914
db365e67adf6
finished with plitting axiom stuff (except that now the module definition needs to be updated)
Kevin Walker <kevin@canyon23.net>
parents:
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diff
changeset
|
693 |
Let $s = \{X_i\}$ be a splitting of X (so $X = \cup_i X_i$). |
db365e67adf6
finished with plitting axiom stuff (except that now the module definition needs to be updated)
Kevin Walker <kevin@canyon23.net>
parents:
913
diff
changeset
|
694 |
Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which restrict to the identity on $\bd X$. |
896
deeff619087e
Initial version of the new splitting axiom.
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diff
changeset
|
695 |
\begin{itemize} |
914
db365e67adf6
finished with plitting axiom stuff (except that now the module definition needs to be updated)
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parents:
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diff
changeset
|
696 |
\item (Alternative 1) Consider the set of homeomorphisms $g:X\to X$ such that $c$ splits along $g(s)$. |
896
deeff619087e
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|
697 |
Then this subset of $\Homeo(X)$ is open and dense. |
914
db365e67adf6
finished with plitting axiom stuff (except that now the module definition needs to be updated)
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parents:
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diff
changeset
|
698 |
Furthermore, if $s$ restricts to a splitting $\bd s$ of $\bd X$, and if $\bd c$ splits along $\bd s$, then the |
db365e67adf6
finished with plitting axiom stuff (except that now the module definition needs to be updated)
Kevin Walker <kevin@canyon23.net>
parents:
913
diff
changeset
|
699 |
intersection of the set of such homeomorphisms $g$ with $\Homeo_\bd(X)$ is open and dense in $\Homeo_\bd(X)$. |
db365e67adf6
finished with plitting axiom stuff (except that now the module definition needs to be updated)
Kevin Walker <kevin@canyon23.net>
parents:
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diff
changeset
|
700 |
\item (Alternative 2) Then there exists an embedded cell complex $S_c \sub X$, called the string locus of $c$, |
db365e67adf6
finished with plitting axiom stuff (except that now the module definition needs to be updated)
Kevin Walker <kevin@canyon23.net>
parents:
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diff
changeset
|
701 |
such that if the splitting $s$ is transverse to $S_c$ then $c$ splits along $s$. |
896
deeff619087e
Initial version of the new splitting axiom.
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diff
changeset
|
702 |
\end{itemize} |
deeff619087e
Initial version of the new splitting axiom.
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diff
changeset
|
703 |
\end{axiom} |
deeff619087e
Initial version of the new splitting axiom.
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diff
changeset
|
704 |
|
deeff619087e
Initial version of the new splitting axiom.
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diff
changeset
|
705 |
We note some consequences of Axiom \ref{axiom:splittings}. |
deeff619087e
Initial version of the new splitting axiom.
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parents:
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diff
changeset
|
706 |
|
deeff619087e
Initial version of the new splitting axiom.
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parents:
892
diff
changeset
|
707 |
First, some preliminary definitions. |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
708 |
If $P$ is a poset let $P\times I$ denote the product poset, where $I = \{0, 1\}$ with ordering $0\le 1$. |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
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diff
changeset
|
709 |
Let $\Cone(P)$ denote $P$ adjoined an additional object $v$ (the vertex of the cone) with $p\le v$ for all objects $p$ of $P$. |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
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diff
changeset
|
710 |
Finally, let $\vcone(P)$ denote $P\times I \cup \Cone(P)$, where we identify $P\times \{0\}$ with the base of the cone. |
deeff619087e
Initial version of the new splitting axiom.
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parents:
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diff
changeset
|
711 |
We call $P\times \{1\}$ the base of $\vcone(P)$. |
deeff619087e
Initial version of the new splitting axiom.
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parents:
892
diff
changeset
|
712 |
(See Figure \ref{vcone-fig}.) |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
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diff
changeset
|
713 |
\begin{figure}[t] |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
714 |
\centering |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
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diff
changeset
|
715 |
\begin{tikzpicture} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
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diff
changeset
|
716 |
[kw node/.style={circle,fill=orange!70}, |
deeff619087e
Initial version of the new splitting axiom.
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parents:
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diff
changeset
|
717 |
kw arrow/.style={-latex, very thick, blue!70, shorten >=.06cm, shorten <=.06cm}, |
deeff619087e
Initial version of the new splitting axiom.
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parents:
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diff
changeset
|
718 |
kw label/.style={cca}, |
deeff619087e
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Kevin Walker <kevin@canyon23.net>
parents:
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diff
changeset
|
719 |
] |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
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diff
changeset
|
720 |
|
deeff619087e
Initial version of the new splitting axiom.
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parents:
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diff
changeset
|
721 |
\definecolor{cca}{rgb}{.1,.4,.3}; |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
722 |
|
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
723 |
\node at (0,0) { |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
724 |
\begin{tikzpicture} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
725 |
\draw |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
726 |
(0,0) node[kw node](p1){} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
727 |
(1,.5) node[kw node](p2){} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
728 |
(2,0) node[kw node](p3){}; |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
729 |
|
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
730 |
\draw[kw arrow] (p1) -- (p3); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
731 |
\draw[kw arrow] (p2) -- (p3); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
732 |
\draw[kw arrow] (p1) -- (p2); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
733 |
|
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
734 |
\draw[kw label] (1,-.6) node{(a)}; |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
735 |
\end{tikzpicture} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
736 |
}; |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
737 |
|
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
738 |
\node at (7,0) { |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
739 |
\begin{tikzpicture} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
740 |
\draw |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
741 |
(0,0) node[kw node](p1){} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
742 |
++(0,2.5) node[kw node](q1){} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
743 |
(1,.5) node[kw node](p2){} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
744 |
++(0,2.5) node[kw node](q2){} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
745 |
(2,0) node[kw node](p3){} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
746 |
++(0,2.5) node[kw node](q3){} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
747 |
; |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
748 |
|
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
749 |
\draw[kw arrow] (p1) -- (p3); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
750 |
\draw[kw arrow] (p2) -- (p3); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
751 |
\draw[kw arrow] (p1) -- (p2); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
752 |
\draw[kw arrow] (q1) -- (q3); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
753 |
\draw[kw arrow] (q2) -- (q3); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
754 |
\draw[kw arrow] (q1) -- (q2); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
755 |
\draw[kw arrow] (p1) -- (q1); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
756 |
\draw[kw arrow] (p2) -- (q2); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
757 |
\draw[kw arrow] (p3) -- (q3); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
758 |
|
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
759 |
\draw[kw label] (1,-.6) node{(b)}; |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
760 |
\end{tikzpicture} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
761 |
}; |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
762 |
|
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
763 |
\node at (0,-5) { |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
764 |
\begin{tikzpicture} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
765 |
\draw |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
766 |
(0,0) node[kw node](p1){} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
767 |
(1,.5) node[kw node](p2){} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
768 |
++(0,2.5) node[kw node](v){} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
769 |
(2,0) node[kw node](p3){} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
770 |
; |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
771 |
|
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
772 |
\draw[kw arrow] (p1) -- (p3); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
773 |
\draw[kw arrow] (p2) -- (p3); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
774 |
\draw[kw arrow] (p1) -- (p2); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
775 |
\draw[kw arrow] (p1) -- (v); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
776 |
\draw[kw arrow] (p2) -- (v); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
777 |
\draw[kw arrow] (p3) -- (v); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
778 |
|
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
779 |
\draw[kw label] (1,-.6) node{(c)}; |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
780 |
\end{tikzpicture} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
781 |
}; |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
782 |
|
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
783 |
\node at (7,-5) { |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
784 |
\begin{tikzpicture} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
785 |
\draw |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
786 |
(0,0) node[kw node](p1){} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
787 |
++(-2,2.5) node[kw node](q1){} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
788 |
(1,.5) node[kw node](p2){} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
789 |
++(-2,2.5) node[kw node](q2){} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
790 |
++(4,0) node[kw node](v){} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
791 |
(2,0) node[kw node](p3){} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
792 |
++(-2,2.5) node[kw node](q3){} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
793 |
; |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
794 |
|
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
795 |
\draw[kw arrow] (p1) -- (p3); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
796 |
\draw[kw arrow] (p2) -- (p3); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
797 |
\draw[kw arrow] (p1) -- (p2); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
798 |
\draw[kw arrow] (p1) -- (v); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
799 |
\draw[kw arrow] (p2) -- (v); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
800 |
\draw[kw arrow] (p3) -- (v); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
801 |
\draw[kw arrow] (q1) -- (q3); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
802 |
\draw[kw arrow] (q2) -- (q3); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
803 |
\draw[kw arrow] (q1) -- (q2); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
804 |
\draw[kw arrow] (p1) -- (q1); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
805 |
\draw[kw arrow] (p2) -- (q2); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
806 |
\draw[kw arrow] (p3) -- (q3); |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
807 |
|
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
808 |
\draw[kw label] (1,-.6) node{(d)}; |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
809 |
\end{tikzpicture} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
810 |
}; |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
811 |
|
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
812 |
\end{tikzpicture} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
813 |
\caption{(a) $P$, (b) $P\times I$, (c) $\Cone(P)$, (d) $\vcone(P)$} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
814 |
\label{vcone-fig} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
815 |
\end{figure} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
816 |
|
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
817 |
|
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
818 |
\begin{lem} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
819 |
\label{lemma:vcones} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
820 |
Let $c\in \cC_k(X)$, with $0\le k < n$, and |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
821 |
let $P$ be a finite poset of splittings of $c$. |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
822 |
Then we can embed $\vcone(P)$ into the splittings of $c$, with $P$ corresponding to the base of $\vcone(P)$. |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
823 |
Furthermore, if $q$ is any decomposition of $X$, then we can take the vertex of $\vcone(P)$ to be $q$ up to a small perturbation. |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
824 |
\end{lem} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
825 |
|
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
826 |
\begin{proof} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
827 |
After a small perturbation, we may assume that $q$ is simultaneously transverse to all the splittings in $P$, and |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
828 |
(by Axiom \ref{axiom:splittings}) that $c$ splits along $q$. |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
829 |
We can now choose, for each splitting $p$ in $P$, a common refinement $p'$ of $p$ and $q$. |
898
14e05e9785c0
minor; unsaved change from a couple of days ago
Kevin Walker <kevin@canyon23.net>
parents:
896
diff
changeset
|
830 |
This constitutes the middle part ($P\times \{0\}$ above) of $\vcone(P)$. |
896
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
831 |
\end{proof} |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
832 |
|
914
db365e67adf6
finished with plitting axiom stuff (except that now the module definition needs to be updated)
Kevin Walker <kevin@canyon23.net>
parents:
913
diff
changeset
|
833 |
\begin{cor} |
db365e67adf6
finished with plitting axiom stuff (except that now the module definition needs to be updated)
Kevin Walker <kevin@canyon23.net>
parents:
913
diff
changeset
|
834 |
For any $c\in \cC_k(X)$, the geometric realization of the poset of splittings of $c$ is contractible. |
db365e67adf6
finished with plitting axiom stuff (except that now the module definition needs to be updated)
Kevin Walker <kevin@canyon23.net>
parents:
913
diff
changeset
|
835 |
\end{cor} |
db365e67adf6
finished with plitting axiom stuff (except that now the module definition needs to be updated)
Kevin Walker <kevin@canyon23.net>
parents:
913
diff
changeset
|
836 |
|
db365e67adf6
finished with plitting axiom stuff (except that now the module definition needs to be updated)
Kevin Walker <kevin@canyon23.net>
parents:
913
diff
changeset
|
837 |
\begin{proof} |
db365e67adf6
finished with plitting axiom stuff (except that now the module definition needs to be updated)
Kevin Walker <kevin@canyon23.net>
parents:
913
diff
changeset
|
838 |
In the geometric realization, V-Cones become actual cones, allowing us to contract any cycle. |
db365e67adf6
finished with plitting axiom stuff (except that now the module definition needs to be updated)
Kevin Walker <kevin@canyon23.net>
parents:
913
diff
changeset
|
839 |
\end{proof} |
db365e67adf6
finished with plitting axiom stuff (except that now the module definition needs to be updated)
Kevin Walker <kevin@canyon23.net>
parents:
913
diff
changeset
|
840 |
|
896
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
841 |
|
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
842 |
\noop{ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
843 |
|
800
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
844 |
We need one additional axiom, in order to constrain the poset of decompositions of a given morphism. |
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
845 |
We will soon want to take colimits (and homotopy colimits) indexed by such posets, and we want to require |
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
846 |
that these colimits are in some sense locally acyclic. |
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
847 |
Before stating the axiom we need a few preliminary definitions. |
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
848 |
If $P$ is a poset let $P\times I$ denote the product poset, where $I = \{0, 1\}$ with ordering $0\le 1$. |
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
849 |
Let $\Cone(P)$ denote $P$ adjoined an additional object $v$ (the vertex of the cone) with $p\le v$ for all objects $p$ of $P$. |
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
850 |
Finally, let $\vcone(P)$ denote $P\times I \cup \Cone(P)$, where we identify $P\times \{0\}$ with the base of the cone. |
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
851 |
We call $P\times \{1\}$ the base of $\vcone(P)$. |
801
33b3e0c065d2
adding placeholder figure
Kevin Walker <kevin@canyon23.net>
parents:
800
diff
changeset
|
852 |
(See Figure \ref{vcone-fig}.) |
33b3e0c065d2
adding placeholder figure
Kevin Walker <kevin@canyon23.net>
parents:
800
diff
changeset
|
853 |
\begin{figure}[t] |
814 | 854 |
\centering |
855 |
\begin{tikzpicture} |
|
856 |
[kw node/.style={circle,fill=orange!70}, |
|
815 | 857 |
kw arrow/.style={-latex, very thick, blue!70, shorten >=.06cm, shorten <=.06cm}, |
814 | 858 |
kw label/.style={cca}, |
859 |
] |
|
860 |
||
861 |
\definecolor{cca}{rgb}{.1,.4,.3}; |
|
862 |
||
863 |
\node at (0,0) { |
|
864 |
\begin{tikzpicture} |
|
865 |
\draw |
|
866 |
(0,0) node[kw node](p1){} |
|
867 |
(1,.5) node[kw node](p2){} |
|
868 |
(2,0) node[kw node](p3){}; |
|
869 |
||
870 |
\draw[kw arrow] (p1) -- (p3); |
|
871 |
\draw[kw arrow] (p2) -- (p3); |
|
872 |
\draw[kw arrow] (p1) -- (p2); |
|
873 |
||
874 |
\draw[kw label] (1,-.6) node{(a)}; |
|
875 |
\end{tikzpicture} |
|
876 |
}; |
|
877 |
||
878 |
\node at (7,0) { |
|
879 |
\begin{tikzpicture} |
|
880 |
\draw |
|
881 |
(0,0) node[kw node](p1){} |
|
882 |
++(0,2.5) node[kw node](q1){} |
|
883 |
(1,.5) node[kw node](p2){} |
|
884 |
++(0,2.5) node[kw node](q2){} |
|
885 |
(2,0) node[kw node](p3){} |
|
886 |
++(0,2.5) node[kw node](q3){} |
|
887 |
; |
|
888 |
||
889 |
\draw[kw arrow] (p1) -- (p3); |
|
890 |
\draw[kw arrow] (p2) -- (p3); |
|
891 |
\draw[kw arrow] (p1) -- (p2); |
|
892 |
\draw[kw arrow] (q1) -- (q3); |
|
893 |
\draw[kw arrow] (q2) -- (q3); |
|
894 |
\draw[kw arrow] (q1) -- (q2); |
|
895 |
\draw[kw arrow] (p1) -- (q1); |
|
896 |
\draw[kw arrow] (p2) -- (q2); |
|
897 |
\draw[kw arrow] (p3) -- (q3); |
|
898 |
||
899 |
\draw[kw label] (1,-.6) node{(b)}; |
|
900 |
\end{tikzpicture} |
|
901 |
}; |
|
902 |
||
903 |
\node at (0,-5) { |
|
904 |
\begin{tikzpicture} |
|
905 |
\draw |
|
906 |
(0,0) node[kw node](p1){} |
|
907 |
(1,.5) node[kw node](p2){} |
|
908 |
++(0,2.5) node[kw node](v){} |
|
909 |
(2,0) node[kw node](p3){} |
|
910 |
; |
|
911 |
||
912 |
\draw[kw arrow] (p1) -- (p3); |
|
913 |
\draw[kw arrow] (p2) -- (p3); |
|
914 |
\draw[kw arrow] (p1) -- (p2); |
|
915 |
\draw[kw arrow] (p1) -- (v); |
|
916 |
\draw[kw arrow] (p2) -- (v); |
|
917 |
\draw[kw arrow] (p3) -- (v); |
|
918 |
||
919 |
\draw[kw label] (1,-.6) node{(c)}; |
|
920 |
\end{tikzpicture} |
|
921 |
}; |
|
922 |
||
923 |
\node at (7,-5) { |
|
924 |
\begin{tikzpicture} |
|
925 |
\draw |
|
926 |
(0,0) node[kw node](p1){} |
|
927 |
++(-2,2.5) node[kw node](q1){} |
|
928 |
(1,.5) node[kw node](p2){} |
|
929 |
++(-2,2.5) node[kw node](q2){} |
|
930 |
++(4,0) node[kw node](v){} |
|
931 |
(2,0) node[kw node](p3){} |
|
932 |
++(-2,2.5) node[kw node](q3){} |
|
933 |
; |
|
934 |
||
935 |
\draw[kw arrow] (p1) -- (p3); |
|
936 |
\draw[kw arrow] (p2) -- (p3); |
|
937 |
\draw[kw arrow] (p1) -- (p2); |
|
938 |
\draw[kw arrow] (p1) -- (v); |
|
939 |
\draw[kw arrow] (p2) -- (v); |
|
940 |
\draw[kw arrow] (p3) -- (v); |
|
941 |
\draw[kw arrow] (q1) -- (q3); |
|
942 |
\draw[kw arrow] (q2) -- (q3); |
|
943 |
\draw[kw arrow] (q1) -- (q2); |
|
944 |
\draw[kw arrow] (p1) -- (q1); |
|
945 |
\draw[kw arrow] (p2) -- (q2); |
|
946 |
\draw[kw arrow] (p3) -- (q3); |
|
947 |
||
948 |
\draw[kw label] (1,-.6) node{(d)}; |
|
949 |
\end{tikzpicture} |
|
950 |
}; |
|
951 |
||
952 |
\end{tikzpicture} |
|
953 |
\caption{(a) $P$, (b) $P\times I$, (c) $\Cone(P)$, (d) $\vcone(P)$} |
|
954 |
\label{vcone-fig} |
|
801
33b3e0c065d2
adding placeholder figure
Kevin Walker <kevin@canyon23.net>
parents:
800
diff
changeset
|
955 |
\end{figure} |
800
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
956 |
|
818
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
957 |
|
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
958 |
\begin{axiom}[Splittings] |
800
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
959 |
\label{axiom:vcones} |
849
cbfbcf204016
no splittability requirement for k=n
Kevin Walker <kevin@canyon23.net>
parents:
837
diff
changeset
|
960 |
Let $c\in \cC_k(X)$, with $0\le k < n$, and |
800
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
961 |
let $P$ be a finite poset of splittings of $c$. |
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
962 |
Then we can embed $\vcone(P)$ into the splittings of $c$, with $P$ corresponding to the base of $\vcone(P)$. |
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
963 |
Furthermore, if $q$ is any decomposition of $X$, then we can take the vertex of $\vcone(P)$ to be $q$ up to a small perturbation. |
818
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
964 |
Also, any splitting of $\bd c$ can be extended to a splitting of $c$. |
800
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
965 |
\end{axiom} |
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
966 |
|
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
967 |
It is easy to see that this axiom holds in our two motivating examples, |
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
968 |
using standard facts about transversality and general position. |
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
969 |
One starts with $q$, perturbs it so that it is in general position with respect to $c$ (in the case of string diagrams) |
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
970 |
and also with respect to each of the decompositions of $P$, then chooses common refinements of each decomposition of $P$ |
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
971 |
and the perturbed $q$. |
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
972 |
These common refinements form the middle ($P\times \{0\}$ above) part of $\vcone(P)$. |
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
973 |
|
805 | 974 |
We note two simple special cases of Axiom \ref{axiom:vcones}. |
800
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
975 |
If $P$ is the empty poset, then $\vcone(P)$ consists of only the vertex, and the axiom says that any morphism $c$ |
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
976 |
can be split along any decomposition of $X$, after a small perturbation. |
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
977 |
If $P$ is the disjoint union of two points, then $\vcone(P)$ looks like a letter W, and the axiom implies that the |
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
978 |
poset of splittings of $c$ is connected. |
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
979 |
Note that we do not require that any two splittings of $c$ have a common refinement (i.e.\ replace the letter W with the letter V). |
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
980 |
Two decompositions of $X$ might intersect in a very messy way, but one can always find a third |
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
981 |
decomposition which has common refinements with each of the original two decompositions. |
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
982 |
|
896
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
983 |
} %%%%%% end \noop %%%%%%%%%%%%%%%%%%%%%%%%%%%% |
800
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
984 |
|
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
985 |
\medskip |
d0b9238aad5d
new n-cat axiom for splittings
Kevin Walker <kevin@canyon23.net>
parents:
799
diff
changeset
|
986 |
|
888
a0fd6e620926
Backed out changeset 7abe7642265e
Kevin Walker <kevin@canyon23.net>
parents:
865
diff
changeset
|
987 |
This completes the definition of an $n$-category. |
a0fd6e620926
Backed out changeset 7abe7642265e
Kevin Walker <kevin@canyon23.net>
parents:
865
diff
changeset
|
988 |
Next we define enriched $n$-categories. |
788
6a1b6c2de201
more reorganization of n-cat defs
Kevin Walker <kevin@canyon23.net>
parents:
787
diff
changeset
|
989 |
|
789
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
parents:
788
diff
changeset
|
990 |
\medskip |
416 | 991 |
|
787
c0cdde54913a
start to rearrange n-cat defs
Kevin Walker <kevin@canyon23.net>
parents:
786
diff
changeset
|
992 |
|
c0cdde54913a
start to rearrange n-cat defs
Kevin Walker <kevin@canyon23.net>
parents:
786
diff
changeset
|
993 |
Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
c0cdde54913a
start to rearrange n-cat defs
Kevin Walker <kevin@canyon23.net>
parents:
786
diff
changeset
|
994 |
The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
789
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
parents:
788
diff
changeset
|
995 |
all $c\in \cl{\cC}(\bd X)$, have the structure of an object in some appropriate auxiliary category |
787
c0cdde54913a
start to rearrange n-cat defs
Kevin Walker <kevin@canyon23.net>
parents:
786
diff
changeset
|
996 |
(e.g.\ vector spaces, or modules over some ring, or chain complexes), |
789
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
parents:
788
diff
changeset
|
997 |
and all the structure maps of the $n$-category are compatible with the auxiliary |
787
c0cdde54913a
start to rearrange n-cat defs
Kevin Walker <kevin@canyon23.net>
parents:
786
diff
changeset
|
998 |
category structure. |
c0cdde54913a
start to rearrange n-cat defs
Kevin Walker <kevin@canyon23.net>
parents:
786
diff
changeset
|
999 |
Note that this auxiliary structure is only in dimension $n$; if $\dim(Y) < n$ then |
c0cdde54913a
start to rearrange n-cat defs
Kevin Walker <kevin@canyon23.net>
parents:
786
diff
changeset
|
1000 |
$\cC(Y; c)$ is just a plain set. |
c0cdde54913a
start to rearrange n-cat defs
Kevin Walker <kevin@canyon23.net>
parents:
786
diff
changeset
|
1001 |
|
795
4d66ffe8dc85
tweak to fam-o-homeo proof; aux enriching cats are sets with extra structure
Kevin Walker <kevin@canyon23.net>
parents:
789
diff
changeset
|
1002 |
%We will aim for a little bit more generality than we need and not assume that the objects |
4d66ffe8dc85
tweak to fam-o-homeo proof; aux enriching cats are sets with extra structure
Kevin Walker <kevin@canyon23.net>
parents:
789
diff
changeset
|
1003 |
%of our auxiliary category are sets with extra structure. |
789
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
parents:
788
diff
changeset
|
1004 |
First we must specify requirements for the auxiliary category. |
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
parents:
788
diff
changeset
|
1005 |
It should have a {\it distributive monoidal structure} in the sense of |
799 | 1006 |
\cite{1010.4527}. |
789
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
parents:
788
diff
changeset
|
1007 |
This means that there is a monoidal structure $\otimes$ and also coproduct $\oplus$, |
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
parents:
788
diff
changeset
|
1008 |
and these two structures interact in the appropriate way. |
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
parents:
788
diff
changeset
|
1009 |
Examples include |
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
parents:
788
diff
changeset
|
1010 |
\begin{itemize} |
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
parents:
788
diff
changeset
|
1011 |
\item vector spaces (or $R$-modules or chain complexes) with tensor product and direct sum; and |
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
parents:
788
diff
changeset
|
1012 |
\item topological spaces with product and disjoint union. |
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
parents:
788
diff
changeset
|
1013 |
\end{itemize} |
795
4d66ffe8dc85
tweak to fam-o-homeo proof; aux enriching cats are sets with extra structure
Kevin Walker <kevin@canyon23.net>
parents:
789
diff
changeset
|
1014 |
For convenience, we will also assume that the objects of our auxiliary category are sets with extra structure. |
4d66ffe8dc85
tweak to fam-o-homeo proof; aux enriching cats are sets with extra structure
Kevin Walker <kevin@canyon23.net>
parents:
789
diff
changeset
|
1015 |
(Otherwise, stating the axioms for identity morphisms becomes more cumbersome.) |
4d66ffe8dc85
tweak to fam-o-homeo proof; aux enriching cats are sets with extra structure
Kevin Walker <kevin@canyon23.net>
parents:
789
diff
changeset
|
1016 |
|
888
a0fd6e620926
Backed out changeset 7abe7642265e
Kevin Walker <kevin@canyon23.net>
parents:
865
diff
changeset
|
1017 |
Before stating the revised axioms for an $n$-category enriched in a distributive monoidal category, |
789
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
parents:
788
diff
changeset
|
1018 |
we need a preliminary definition. |
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
parents:
788
diff
changeset
|
1019 |
Once we have the above $n$-category axioms for $n{-}1$-morphisms, we can define the |
787914e9e859
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|
1020 |
category $\bbc$ of {\it $n$-balls with boundary conditions}. |
787914e9e859
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788
diff
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|
1021 |
Its objects are pairs $(X, c)$, where $X$ is an $n$-ball and $c \in \cl\cC(\bd X)$ is the ``boundary condition". |
897
9ba67422f1b9
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896
diff
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|
1022 |
The morphisms from $(X, c)$ to $(X', c')$, denoted $\Homeo(X; c \to X'; c')$, are |
797
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796
diff
changeset
|
1023 |
homeomorphisms $f:X\to X'$ such that $f|_{\bd X}(c) = c'$. |
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796
diff
changeset
|
1024 |
%Let $\pi_0(\bbc)$ denote |
789
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
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788
diff
changeset
|
1025 |
|
888
a0fd6e620926
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|
1026 |
\begin{axiom}[Enriched $n$-categories] |
789
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
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788
diff
changeset
|
1027 |
\label{axiom:enriched} |
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
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788
diff
changeset
|
1028 |
Let $\cS$ be a distributive symmetric monoidal category. |
888
a0fd6e620926
Backed out changeset 7abe7642265e
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865
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changeset
|
1029 |
An $n$-category enriched in $\cS$ satisfies the above $n$-category axioms for $k=0,\ldots,n-1$, |
789
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
parents:
788
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changeset
|
1030 |
and modifies the axioms for $k=n$ as follows: |
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
parents:
788
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changeset
|
1031 |
\begin{itemize} |
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
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788
diff
changeset
|
1032 |
\item Morphisms. We have a functor $\cC_n$ from $\bbc$ ($n$-balls with boundary conditions) to $\cS$. |
797
40729de8e067
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796
diff
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|
1033 |
%[already said this above. ack] Furthermore, $\cC_n(f)$ depends only on the path component of a homeomorphism $f: (X, c) \to (X', c')$. |
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796
diff
changeset
|
1034 |
%In particular, homeomorphisms which are isotopic to the identity rel boundary act trivially |
789
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
parents:
788
diff
changeset
|
1035 |
\item Composition. Let $B = B_1\cup_Y B_2$ as in Axiom \ref{axiom:composition}. |
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
parents:
788
diff
changeset
|
1036 |
Let $Y_i = \bd B_i \setmin Y$. |
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
parents:
788
diff
changeset
|
1037 |
Note that $\bd B = Y_1\cup Y_2$. |
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
parents:
788
diff
changeset
|
1038 |
Let $c_i \in \cC(Y_i)$ with $\bd c_1 = \bd c_2 = d \in \cl\cC(E)$. |
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
parents:
788
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|
1039 |
Then we have a map |
787914e9e859
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Kevin Walker <kevin@canyon23.net>
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788
diff
changeset
|
1040 |
\[ |
787914e9e859
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Kevin Walker <kevin@canyon23.net>
parents:
788
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|
1041 |
\gl_Y : \bigoplus_c \cC(B_1; c_1 \bullet c) \otimes \cC(B_2; c_2\bullet c) \to \cC(B; c_1\bullet c_2), |
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
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788
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changeset
|
1042 |
\] |
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
parents:
788
diff
changeset
|
1043 |
where the sum is over $c\in\cC(Y)$ such that $\bd c = d$. |
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
parents:
788
diff
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|
1044 |
This map is natural with respect to the action of homeomorphisms and with respect to restrictions. |
795
4d66ffe8dc85
tweak to fam-o-homeo proof; aux enriching cats are sets with extra structure
Kevin Walker <kevin@canyon23.net>
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789
diff
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|
1045 |
%\item Product morphisms. \nn{Hmm... not sure what to say here. maybe we need sets with structure after all.} |
789
787914e9e859
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788
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|
1046 |
\end{itemize} |
787914e9e859
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Kevin Walker <kevin@canyon23.net>
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788
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|
1047 |
\end{axiom} |
787914e9e859
axioms for enriched n-cats; but these might need to be modified since the product axiom seems to require that these are sets with structure after all
Kevin Walker <kevin@canyon23.net>
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788
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changeset
|
1048 |
|
796
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Kevin Walker <kevin@canyon23.net>
parents:
795
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changeset
|
1049 |
\medskip |
789
787914e9e859
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Kevin Walker <kevin@canyon23.net>
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788
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changeset
|
1050 |
|
796
d30537de52c7
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795
diff
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|
1051 |
When the enriching category $\cS$ is chain complexes or topological spaces, |
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795
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|
1052 |
or more generally an appropriate sort of $\infty$-category, |
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795
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|
1053 |
we can modify the extended isotopy axiom \ref{axiom:extended-isotopies} |
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|
1054 |
to require that families of homeomorphisms act |
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a0fd6e620926
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|
1055 |
and obtain what we shall call an $A_\infty$ $n$-category. |
787
c0cdde54913a
start to rearrange n-cat defs
Kevin Walker <kevin@canyon23.net>
parents:
786
diff
changeset
|
1056 |
|
797
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diff
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|
1057 |
\noop{ |
796
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|
1058 |
We believe that abstract definitions should be guided by diverse collections |
d30537de52c7
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Kevin Walker <kevin@canyon23.net>
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diff
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|
1059 |
of concrete examples, and a lack of diversity in our present collection of examples of $A_\infty$ $n$-categories |
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|
1060 |
makes us reluctant to commit to an all-encompassing general definition. |
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|
1061 |
Instead, we will give a relatively narrow definition which covers the examples we consider in this paper. |
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|
1062 |
After stating it, we will briefly discuss ways in which it can be made more general. |
797
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|
1063 |
} |
788
6a1b6c2de201
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Kevin Walker <kevin@canyon23.net>
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diff
changeset
|
1064 |
|
797
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changeset
|
1065 |
Recall the category $\bbc$ of balls with boundary conditions. |
897
9ba67422f1b9
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Scott Morrison <scott@tqft.net>
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|
1066 |
Note that the morphisms $\Homeo(X;c \to X'; c')$ from $(X, c)$ to $(X', c')$ form a topological space. |
797
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|
1067 |
Let $\cS$ be an appropriate $\infty$-category (e.g.\ chain complexes) |
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|
1068 |
and let $\cJ$ be an $\infty$-functor from topological spaces to $\cS$ |
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|
1069 |
(e.g.\ the singular chain functor $C_*$). |
788
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|
1070 |
|
833 | 1071 |
\begin{axiom}[\textup{\textbf{[$A_\infty$ replacement for Axiom \ref{axiom:extended-isotopies}]}} Families of homeomorphisms act in dimension $n$.] |
560 | 1072 |
\label{axiom:families} |
797
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|
1073 |
For each pair of $n$-balls $X$ and $X'$ and each pair $c\in \cl{\cC}(\bd X)$ and $c'\in \cl{\cC}(\bd X')$ we have an $\cS$-morphism |
97 | 1074 |
\[ |
897
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diff
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|
1075 |
\cJ(\Homeo(X;c \to X'; c')) \ot \cC(X; c) \to \cC(X'; c') . |
97 | 1076 |
\] |
797
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|
1077 |
Similarly, we have an $\cS$-morphism |
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diff
changeset
|
1078 |
\[ |
40729de8e067
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Kevin Walker <kevin@canyon23.net>
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diff
changeset
|
1079 |
\cJ(\Coll(X,c)) \ot \cC(X; c) \to \cC(X; c), |
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diff
changeset
|
1080 |
\] |
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changeset
|
1081 |
where $\Coll(X,c)$ denotes the space of collar maps. |
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diff
changeset
|
1082 |
(See below for further discussion.) |
796
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changeset
|
1083 |
These action maps are required to be associative up to coherent homotopy, |
475
07c18e2abd8f
redefine "permissible decomp", and other changes to ntcat.tex; should be read
Kevin Walker <kevin@canyon23.net>
parents:
463
diff
changeset
|
1084 |
and also compatible with composition (gluing) in the sense that |
437 | 1085 |
a diagram like the one in Theorem \ref{thm:CH} commutes. |
797
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|
1086 |
% say something about compatibility with product morphisms? |
266
e2bab777d7c9
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Scott Morrison <scott@tqft.net>
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265
diff
changeset
|
1087 |
\end{axiom} |
97 | 1088 |
|
797
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|
1089 |
We now describe the topology on $\Coll(X; c)$. |
897
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|
1090 |
We retain notation from the above definition of collar map (after Axiom \ref{axiom:isotopy-preliminary}). |
797
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796
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changeset
|
1091 |
Each collaring homeomorphism $X \cup (Y\times J) \to X$ determines a map from points $p$ of $\bd X$ to |
833 | 1092 |
(possibly length zero) embedded intervals in $X$ terminating at $p$. |
797
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Kevin Walker <kevin@canyon23.net>
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changeset
|
1093 |
If $p \in Y$ this interval is the image of $\{p\}\times J$. |
833 | 1094 |
If $p \notin Y$ then $p$ is assigned the length zero interval $\{p\}$. |
797
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Kevin Walker <kevin@canyon23.net>
parents:
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diff
changeset
|
1095 |
Such collections of intervals have a natural topology, and $\Coll(X; c)$ inherits its topology from this. |
833 | 1096 |
Note in particular that parts of the collar are allowed to shrink continuously to zero length. |
1097 |
(This is the real content; if nothing shrinks to zero length then the action of families of collar |
|
797
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|
1098 |
maps follows from the action of families of homeomorphisms and compatibility with gluing.) |
97 | 1099 |
|
797
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|
1100 |
The $k=n$ case of Axiom \ref{axiom:morphisms} posits a {\it strictly} associative action of {\it sets} |
897
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changeset
|
1101 |
$\Homeo(X;c\to X'; c') \times \cC(X; c) \to \cC(X'; c')$, and at first it might seem that this would force the above |
9ba67422f1b9
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Scott Morrison <scott@tqft.net>
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changeset
|
1102 |
action of $\cJ(\Homeo(X;c\to X'; c'))$ to be strictly associative as well (assuming the two actions are compatible). |
797
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|
1103 |
In fact, compatibility implies less than this. |
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|
1104 |
For simplicity, assume that $\cJ$ is $C_*$, the singular chains functor. |
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|
1105 |
(This is the example most relevant to this paper.) |
897
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|
1106 |
Then compatibility implies that the action of $C_*(\Homeo(X;c\to X'; c'))$ agrees with the action |
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changeset
|
1107 |
of $C_0(\Homeo(X;c\to X'; c'))$ coming from Axiom \ref{axiom:morphisms}, so we only require associativity in degree zero. |
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|
1108 |
And indeed, this is true for our main example of an $A_\infty$ $n$-category based on the blob construction (see Example \ref{ex:blob-complexes-of-balls} below). |
797
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|
1109 |
Stating this sort of compatibility for general $\cS$ and $\cJ$ requires further assumptions, |
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changeset
|
1110 |
such as the forgetful functor from $\cS$ to sets having a left adjoint, and $\cS$ having an internal Hom. |
821
6868130229bf
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Kevin Walker <kevin@canyon23.net>
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820
diff
changeset
|
1111 |
|
797
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changeset
|
1112 |
An alternative (due to Peter Teichner) is to say that Axiom \ref{axiom:families} |
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changeset
|
1113 |
supersedes the $k=n$ case of Axiom \ref{axiom:morphisms}; in dimension $n$ we just have a |
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changeset
|
1114 |
functor $\bbc \to \cS$ of $A_\infty$ 1-categories. |
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|
1115 |
(This assumes some prior notion of $A_\infty$ 1-category.) |
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|
1116 |
We are not currently aware of any examples which require this sort of greater generality, so we think it best |
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|
1117 |
to refrain from settling on a preferred version of the axiom until |
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|
1118 |
we have a greater variety of examples to guide the choice. |
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|
1119 |
|
822
9e695fc9b13c
add remark about a-inf axiom implying isotopy invariance
Kevin Walker <kevin@canyon23.net>
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821
diff
changeset
|
1120 |
Note that if we think of an ordinary 1-category as an $A_\infty$ 1-category where $k$-morphisms are identities for $k>1$, |
9e695fc9b13c
add remark about a-inf axiom implying isotopy invariance
Kevin Walker <kevin@canyon23.net>
parents:
821
diff
changeset
|
1121 |
then Axiom \ref{axiom:families} implies Axiom \ref{axiom:extended-isotopies}. |
821
6868130229bf
minor; out of time for now
Kevin Walker <kevin@canyon23.net>
parents:
820
diff
changeset
|
1122 |
|
797
40729de8e067
finish fam-o-homeo axiom revisions and discussion
Kevin Walker <kevin@canyon23.net>
parents:
796
diff
changeset
|
1123 |
Another variant of the above axiom would be to drop the ``up to homotopy" and require a strictly associative action. |
853
870d6fac5420
several minor corrections, from referee
Scott Morrison <scott@tqft.net>
parents:
850
diff
changeset
|
1124 |
In fact, the alternative construction $\btc_*(X)$ of the blob complex described in \S \ref{ss:alt-def} |
861
84bb5ab4c85c
unfinished edits to fam-o-homeo lemma and EB_n algebra example
Kevin Walker <kevin@canyon23.net>
parents:
859
diff
changeset
|
1125 |
gives $n$-categories as in Example \ref{ex:blob-complexes-of-balls} which satisfy this stronger axiom. |
84bb5ab4c85c
unfinished edits to fam-o-homeo lemma and EB_n algebra example
Kevin Walker <kevin@canyon23.net>
parents:
859
diff
changeset
|
1126 |
%since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across. |
84bb5ab4c85c
unfinished edits to fam-o-homeo lemma and EB_n algebra example
Kevin Walker <kevin@canyon23.net>
parents:
859
diff
changeset
|
1127 |
For future reference we make the following definition. |
84bb5ab4c85c
unfinished edits to fam-o-homeo lemma and EB_n algebra example
Kevin Walker <kevin@canyon23.net>
parents:
859
diff
changeset
|
1128 |
|
84bb5ab4c85c
unfinished edits to fam-o-homeo lemma and EB_n algebra example
Kevin Walker <kevin@canyon23.net>
parents:
859
diff
changeset
|
1129 |
\begin{defn} |
888
a0fd6e620926
Backed out changeset 7abe7642265e
Kevin Walker <kevin@canyon23.net>
parents:
865
diff
changeset
|
1130 |
A {\em strict $A_\infty$ $n$-category} is one in which the actions of Axiom \ref{axiom:families} are strictly associative. |
861
84bb5ab4c85c
unfinished edits to fam-o-homeo lemma and EB_n algebra example
Kevin Walker <kevin@canyon23.net>
parents:
859
diff
changeset
|
1131 |
\end{defn} |
679
72a1d5014abc
compatibility of first and last n-cat axioms; mention stricter variant of last axiom
Kevin Walker <kevin@canyon23.net>
parents:
611
diff
changeset
|
1132 |
|
797
40729de8e067
finish fam-o-homeo axiom revisions and discussion
Kevin Walker <kevin@canyon23.net>
parents:
796
diff
changeset
|
1133 |
\noop{ |
103 | 1134 |
Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
680
0591d017e698
plain n-cat -> ordinary n-cat
Kevin Walker <kevin@canyon23.net>
parents:
679
diff
changeset
|
1135 |
into a ordinary $n$-category (enriched over graded groups). |
266
e2bab777d7c9
minor changes, fixes to some diagrams
Scott Morrison <scott@tqft.net>
parents:
265
diff
changeset
|
1136 |
In a different direction, if we enrich over topological spaces instead of chain complexes, |
97 | 1137 |
we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
1138 |
instead of $C_*(\Homeo_\bd(X))$. |
|
266
e2bab777d7c9
minor changes, fixes to some diagrams
Scott Morrison <scott@tqft.net>
parents:
265
diff
changeset
|
1139 |
Taking singular chains converts such a space type $A_\infty$ $n$-category into a chain complex |
97 | 1140 |
type $A_\infty$ $n$-category. |
797
40729de8e067
finish fam-o-homeo axiom revisions and discussion
Kevin Walker <kevin@canyon23.net>
parents:
796
diff
changeset
|
1141 |
} |
796
d30537de52c7
in the midst of revising a-inf and enriched n-cat axioms; not done yet
Kevin Walker <kevin@canyon23.net>
parents:
795
diff
changeset
|
1142 |
|
d30537de52c7
in the midst of revising a-inf and enriched n-cat axioms; not done yet
Kevin Walker <kevin@canyon23.net>
parents:
795
diff
changeset
|
1143 |
|
99 | 1144 |
\medskip |
97 | 1145 |
|
888
a0fd6e620926
Backed out changeset 7abe7642265e
Kevin Walker <kevin@canyon23.net>
parents:
865
diff
changeset
|
1146 |
We define a $j$ times monoidal $n$-category to be an $(n{+}j)$-category $\cC$ where |
750
4b1f08238bae
added brief def of monoidal n-cats; killed some old invisible comments
Kevin Walker <kevin@canyon23.net>
parents:
741
diff
changeset
|
1147 |
$\cC(X)$ is a trivial 1-element set if $X$ is a $k$-ball with $k<j$. |
4b1f08238bae
added brief def of monoidal n-cats; killed some old invisible comments
Kevin Walker <kevin@canyon23.net>
parents:
741
diff
changeset
|
1148 |
See Example \ref{ex:bord-cat}. |
4b1f08238bae
added brief def of monoidal n-cats; killed some old invisible comments
Kevin Walker <kevin@canyon23.net>
parents:
741
diff
changeset
|
1149 |
|
4b1f08238bae
added brief def of monoidal n-cats; killed some old invisible comments
Kevin Walker <kevin@canyon23.net>
parents:
741
diff
changeset
|
1150 |
\medskip |
4b1f08238bae
added brief def of monoidal n-cats; killed some old invisible comments
Kevin Walker <kevin@canyon23.net>
parents:
741
diff
changeset
|
1151 |
|
888
a0fd6e620926
Backed out changeset 7abe7642265e
Kevin Walker <kevin@canyon23.net>
parents:
865
diff
changeset
|
1152 |
The alert reader will have already noticed that our definition of an (ordinary) $n$-category |
416 | 1153 |
is extremely similar to our definition of a system of fields. |
1154 |
There are two differences. |
|
329
eb03c4a92f98
various changes, mostly rewriting intros to sections for exposition
Scott Morrison <scott@tqft.net>
parents:
328
diff
changeset
|
1155 |
First, for the $n$-category definition we restrict our attention to balls |
99 | 1156 |
(and their boundaries), while for fields we consider all manifolds. |
885
61541264d4b3
finishing most of the minor/typo issues from the referee
Scott Morrison <scott@tqft.net>
parents:
882
diff
changeset
|
1157 |
Second, in the category definition we directly impose isotopy |
416 | 1158 |
invariance in dimension $n$, while in the fields definition we |
1159 |
instead remember a subspace of local relations which contain differences of isotopic fields. |
|
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1160 |
(Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.) |
900
2efd26072c91
more referee comments, mostly adding cross-references to examples, lemma-izing the fields -> ncats construction
Scott Morrison <scott@tqft.net>
parents:
897
diff
changeset
|
1161 |
Thus |
2efd26072c91
more referee comments, mostly adding cross-references to examples, lemma-izing the fields -> ncats construction
Scott Morrison <scott@tqft.net>
parents:
897
diff
changeset
|
1162 |
\begin{lem} |
2efd26072c91
more referee comments, mostly adding cross-references to examples, lemma-izing the fields -> ncats construction
Scott Morrison <scott@tqft.net>
parents:
897
diff
changeset
|
1163 |
\label{lem:ncat-from-fields} |
2efd26072c91
more referee comments, mostly adding cross-references to examples, lemma-izing the fields -> ncats construction
Scott Morrison <scott@tqft.net>
parents:
897
diff
changeset
|
1164 |
A system of fields and local relations $(\cF,U)$ determines an $n$-category $\cC_ {\cF,U}$ simply by restricting our attention to |
329
eb03c4a92f98
various changes, mostly rewriting intros to sections for exposition
Scott Morrison <scott@tqft.net>
parents:
328
diff
changeset
|
1165 |
balls and, at level $n$, quotienting out by the local relations: |
eb03c4a92f98
various changes, mostly rewriting intros to sections for exposition
Scott Morrison <scott@tqft.net>
parents:
328
diff
changeset
|
1166 |
\begin{align*} |
494
cb76847c439e
many small fixes in ncat.tex
Scott Morrison <scott@tqft.net>
parents:
479
diff
changeset
|
1167 |
\cC_{\cF,U}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / U(B) & \text{when $k=n$.}\end{cases} |
329
eb03c4a92f98
various changes, mostly rewriting intros to sections for exposition
Scott Morrison <scott@tqft.net>
parents:
328
diff
changeset
|
1168 |
\end{align*} |
900
2efd26072c91
more referee comments, mostly adding cross-references to examples, lemma-izing the fields -> ncats construction
Scott Morrison <scott@tqft.net>
parents:
897
diff
changeset
|
1169 |
\end{lem} |
142 | 1170 |
This $n$-category can be thought of as the local part of the fields. |
685
8efbd2730ef9
"topological n-cat" --> either "disk-like n-cat" or "ordinary n-cat" (when contrasted with A-inf n-cat)
Kevin Walker <kevin@canyon23.net>
parents:
683
diff
changeset
|
1171 |
Conversely, given a disk-like $n$-category we can construct a system of fields via |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1172 |
a colimit construction; see \S \ref{ss:ncat_fields} below. |
99 | 1173 |
|
850 | 1174 |
\medskip |
1175 |
||
682
5f22b4501e5f
summary of data and properties for n-cats
Kevin Walker <kevin@canyon23.net>
parents:
680
diff
changeset
|
1176 |
In the $n$-category axioms above we have intermingled data and properties for expository reasons. |
5f22b4501e5f
summary of data and properties for n-cats
Kevin Walker <kevin@canyon23.net>
parents:
680
diff
changeset
|
1177 |
Here's a summary of the definition which segregates the data from the properties. |
887
ab0b4827c89c
more referee report stuff, relatively minor
Kevin Walker <kevin@canyon23.net>
parents:
885
diff
changeset
|
1178 |
We also remind the reader of the inductive nature of the definition: All the data for $k{-}1$-morphisms must be in place |
ab0b4827c89c
more referee report stuff, relatively minor
Kevin Walker <kevin@canyon23.net>
parents:
885
diff
changeset
|
1179 |
before we can describe the data for $k$-morphisms. |
682
5f22b4501e5f
summary of data and properties for n-cats
Kevin Walker <kevin@canyon23.net>
parents:
680
diff
changeset
|
1180 |
|
888
a0fd6e620926
Backed out changeset 7abe7642265e
Kevin Walker <kevin@canyon23.net>
parents:
865
diff
changeset
|
1181 |
An $n$-category consists of the following data: |
682
5f22b4501e5f
summary of data and properties for n-cats
Kevin Walker <kevin@canyon23.net>
parents:
680
diff
changeset
|
1182 |
\begin{itemize} |
689
5ab2b1b2c9db
trying out a semicolon list
Scott Morrison <scott@tqft.net>
parents:
688
diff
changeset
|
1183 |
\item functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms}); |
5ab2b1b2c9db
trying out a semicolon list
Scott Morrison <scott@tqft.net>
parents:
688
diff
changeset
|
1184 |
\item boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary}); |
727
0ec80a7773dc
added two more transverse symbols
Kevin Walker <kevin@canyon23.net>
parents:
726
diff
changeset
|
1185 |
\item ``composition'' or ``gluing'' maps $\gl_Y : \cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E \to \cC(B_1\cup_Y B_2)\trans E$ (Axiom \ref{axiom:composition}); |
689
5ab2b1b2c9db
trying out a semicolon list
Scott Morrison <scott@tqft.net>
parents:
688
diff
changeset
|
1186 |
\item ``product'' or ``identity'' maps $\pi^*:\cC(X)\to \cC(E)$ for each pinched product $\pi:E\to X$ (Axiom \ref{axiom:product}); |
820
57425531f564
update n-cat summary lists
Kevin Walker <kevin@canyon23.net>
parents:
818
diff
changeset
|
1187 |
\item if enriching in an auxiliary category, additional structure on $\cC_n(X; c)$ (Axiom \ref{axiom:enriched}); |
57425531f564
update n-cat summary lists
Kevin Walker <kevin@canyon23.net>
parents:
818
diff
changeset
|
1188 |
%\item in the $A_\infty$ case, an action of $C_*(\Homeo_\bd(X))$, and similarly for families of collar maps (Axiom \ref{axiom:families}). |
57425531f564
update n-cat summary lists
Kevin Walker <kevin@canyon23.net>
parents:
818
diff
changeset
|
1189 |
\item in the $A_\infty$ case, actions of the topological spaces of homeomorphisms preserving boundary conditions |
57425531f564
update n-cat summary lists
Kevin Walker <kevin@canyon23.net>
parents:
818
diff
changeset
|
1190 |
and collar maps (Axiom \ref{axiom:families}). |
682
5f22b4501e5f
summary of data and properties for n-cats
Kevin Walker <kevin@canyon23.net>
parents:
680
diff
changeset
|
1191 |
\end{itemize} |
837 | 1192 |
The above data must satisfy the following conditions. |
682
5f22b4501e5f
summary of data and properties for n-cats
Kevin Walker <kevin@canyon23.net>
parents:
680
diff
changeset
|
1193 |
\begin{itemize} |
5f22b4501e5f
summary of data and properties for n-cats
Kevin Walker <kevin@canyon23.net>
parents:
680
diff
changeset
|
1194 |
\item The gluing maps are compatible with actions of homeomorphisms and boundary |
5f22b4501e5f
summary of data and properties for n-cats
Kevin Walker <kevin@canyon23.net>
parents:
680
diff
changeset
|
1195 |
restrictions (Axiom \ref{axiom:composition}). |
5f22b4501e5f
summary of data and properties for n-cats
Kevin Walker <kevin@canyon23.net>
parents:
680
diff
changeset
|
1196 |
\item For $k<n$ the gluing maps are injective (Axiom \ref{axiom:composition}). |
5f22b4501e5f
summary of data and properties for n-cats
Kevin Walker <kevin@canyon23.net>
parents:
680
diff
changeset
|
1197 |
\item The gluing maps are strictly associative (Axiom \ref{nca-assoc}). |
5f22b4501e5f
summary of data and properties for n-cats
Kevin Walker <kevin@canyon23.net>
parents:
680
diff
changeset
|
1198 |
\item The product maps are associative and also compatible with homeomorphism actions, gluing and restriction (Axiom \ref{axiom:product}). |
5f22b4501e5f
summary of data and properties for n-cats
Kevin Walker <kevin@canyon23.net>
parents:
680
diff
changeset
|
1199 |
\item If enriching in an auxiliary category, all of the data should be compatible |
820
57425531f564
update n-cat summary lists
Kevin Walker <kevin@canyon23.net>
parents:
818
diff
changeset
|
1200 |
with the auxiliary category structure on $\cC_n(X; c)$ (Axiom \ref{axiom:enriched}). |
896
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
1201 |
\item The possible splittings of a morphism satisfy various conditions (Axiom \ref{axiom:splittings}). |
820
57425531f564
update n-cat summary lists
Kevin Walker <kevin@canyon23.net>
parents:
818
diff
changeset
|
1202 |
\item For ordinary categories, invariance of $n$-morphisms under extended isotopies |
57425531f564
update n-cat summary lists
Kevin Walker <kevin@canyon23.net>
parents:
818
diff
changeset
|
1203 |
and collar maps (Axiom \ref{axiom:extended-isotopies}). |
682
5f22b4501e5f
summary of data and properties for n-cats
Kevin Walker <kevin@canyon23.net>
parents:
680
diff
changeset
|
1204 |
\end{itemize} |
5f22b4501e5f
summary of data and properties for n-cats
Kevin Walker <kevin@canyon23.net>
parents:
680
diff
changeset
|
1205 |
|
5f22b4501e5f
summary of data and properties for n-cats
Kevin Walker <kevin@canyon23.net>
parents:
680
diff
changeset
|
1206 |
|
888
a0fd6e620926
Backed out changeset 7abe7642265e
Kevin Walker <kevin@canyon23.net>
parents:
865
diff
changeset
|
1207 |
\subsection{Examples of \texorpdfstring{$n$}{n}-categories} |
309
386d2d12f95b
start E_n example; other minor changes
Kevin Walker <kevin@canyon23.net>
parents:
303
diff
changeset
|
1208 |
\label{ss:ncat-examples} |
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
1209 |
|
101 | 1210 |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1211 |
We now describe several classes of examples of $n$-categories satisfying our axioms. |
418
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
1212 |
We typically specify only the morphisms; the rest of the data for the category |
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
1213 |
(restriction maps, gluing, product morphisms, action of homeomorphisms) is usually obvious. |
101 | 1214 |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1215 |
\begin{example}[Maps to a space] |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1216 |
\rm |
190
16efb5711c6f
minor edits in ncats
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
189
diff
changeset
|
1217 |
\label{ex:maps-to-a-space}% |
425 | 1218 |
Let $T$ be a topological space. |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1219 |
We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows. |
310
ee7be19ee61a
converting sphere axiom to a proposition; still need to make similar changes in module axioms
Kevin Walker <kevin@canyon23.net>
parents:
309
diff
changeset
|
1220 |
For $X$ a $k$-ball with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1221 |
all continuous maps from $X$ to $T$. |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1222 |
For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo |
196 | 1223 |
homotopies fixed on $\bd X$. |
101 | 1224 |
(Note that homotopy invariance implies isotopy invariance.) |
1225 |
For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
|
1226 |
be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. |
|
418
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
1227 |
\end{example} |
313 | 1228 |
|
418
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
1229 |
\noop{ |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1230 |
Recall we described a system of fields and local relations based on maps to $T$ in Example \ref{ex:maps-to-a-space(fields)} above. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1231 |
Constructing a system of fields from $\pi_{\leq n}(T)$ recovers that example. |
418
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
1232 |
\nn{shouldn't this go elsewhere? we haven't yet discussed constructing a system of fields from |
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
1233 |
an n-cat} |
a96f3d2ef852
revisions of n-cat examples
Kevin Walker <kevin@canyon23.net>
parents:
417
diff
changeset
|
1234 |
} |
101 | 1235 |
|
423 | 1236 |
\begin{example}[Maps to a space, with a fiber] \label{ex:maps-with-fiber} |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1237 |
\rm |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1238 |
\label{ex:maps-to-a-space-with-a-fiber}% |
196 | 1239 |
We can modify the example above, by fixing a |
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1240 |
closed $m$-manifold $F$, and defining $\pi^{\times F}_{\leq n}(T)(X) = \Maps(X \times F \to T)$, |
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1241 |
otherwise leaving the definition in Example \ref{ex:maps-to-a-space} unchanged. |
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1242 |
Taking $F$ to be a point recovers the previous case. |
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1243 |
\end{example} |
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1244 |
|
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1245 |
\begin{example}[Linearized, twisted, maps to a space] |
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1246 |
\rm |
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|
1247 |
\label{ex:linearized-maps-to-a-space}% |
191
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|
1248 |
We can linearize Examples \ref{ex:maps-to-a-space} and \ref{ex:maps-to-a-space-with-a-fiber} as follows. |
101 | 1249 |
Let $\alpha$ be an $(n{+}m{+}1)$-cocycle on $T$ with values in a ring $R$ |
191
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|
1250 |
(have in mind the trivial cocycle). |
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|
1251 |
For $X$ of dimension less than $n$ define $\pi^{\alpha, \times F}_{\leq n}(T)(X)$ as before, ignoring $\alpha$. |
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|
1252 |
For $X$ an $n$-ball and $c\in \Maps(\bdy X \times F \to T)$ define $\pi^{\alpha, \times F}_{\leq n}(T)(X; c)$ to be |
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|
1253 |
the $R$-module of finite linear combinations of continuous maps from $X\times F$ to $T$, |
101 | 1254 |
modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy |
191
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1255 |
$h: X\times F\times I \to T$, then $a = \alpha(h)b$. |
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|
1256 |
(In order for this to be well-defined we must choose $\alpha$ to be zero on degenerate simplices. |
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|
1257 |
Alternatively, we could equip the balls with fundamental classes.) |
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|
1258 |
\end{example} |
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1259 |
|
425 | 1260 |
\begin{example}[$n$-categories from TQFTs] |
1261 |
\rm |
|
1262 |
\label{ex:ncats-from-tqfts}% |
|
1263 |
Let $\cF$ be a TQFT in the sense of \S\ref{sec:fields}: an $n$-dimensional |
|
1264 |
system of fields (also denoted $\cF$) and local relations. |
|
1265 |
Let $W$ be an $n{-}j$-manifold. |
|
1266 |
Define the $j$-category $\cF(W)$ as follows. |
|
1267 |
If $X$ is a $k$-ball with $k<j$, let $\cF(W)(X) \deq \cF(W\times X)$. |
|
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|
1268 |
If $X$ is a $j$-ball and $c\in \cl{\cF(W)}(\bd X)$, |
425 | 1269 |
let $\cF(W)(X; c) \deq A_\cF(W\times X; c)$. |
1270 |
\end{example} |
|
1271 |
||
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|
1272 |
This last example generalizes Lemma \ref{lem:ncat-from-fields} above which produced an $n$-category from an $n$-dimensional system of fields and local relations. Taking $W$ to be the point recovers that statement. |
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|
1273 |
|
425 | 1274 |
The next example is only intended to be illustrative, as we don't specify |
1275 |
which definition of a ``traditional $n$-category" we intend. |
|
1276 |
Further, most of these definitions don't even have an agreed-upon notion of |
|
1277 |
``strong duality", which we assume here. |
|
191
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|
1278 |
\begin{example}[Traditional $n$-categories] |
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|
1279 |
\rm |
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|
1280 |
\label{ex:traditional-n-categories} |
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|
1281 |
Given a ``traditional $n$-category with strong duality" $C$ |
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|
1282 |
define $\cC(X)$, for $X$ a $k$-ball with $k < n$, |
346
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|
1283 |
to be the set of all $C$-labeled embedded cell complexes of $X$ (c.f. \S \ref{sec:fields}). |
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|
1284 |
For $X$ an $n$-ball and $c\in \cl{\cC}(\bd X)$, define $\cC(X; c)$ to be finite linear |
346
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|
1285 |
combinations of $C$-labeled embedded cell complexes of $X$ |
191
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|
1286 |
modulo the kernel of the evaluation map. |
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|
1287 |
Define a product morphism $a\times D$, for $D$ an $m$-ball, to be the product of the cell complex of $a$ with $D$, |
346
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|
1288 |
with each cell labelled according to the corresponding cell for $a$. |
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|
1289 |
(These two cells have the same codimension.) |
191
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|
1290 |
More generally, start with an $n{+}m$-category $C$ and a closed $m$-manifold $F$. |
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|
1291 |
Define $\cC(X)$, for $\dim(X) < n$, |
346
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|
1292 |
to be the set of all $C$-labeled embedded cell complexes of $X\times F$. |
191
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|
1293 |
Define $\cC(X; c)$, for $X$ an $n$-ball, |
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|
1294 |
to be the dual Hilbert space $A(X\times F; c)$. |
426
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|
1295 |
(See \S\ref{sec:constructing-a-tqft}.) |
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|
1296 |
\end{example} |
313 | 1297 |
|
204 | 1298 |
|
775 | 1299 |
\begin{example}[The bordism $n$-category of $d$-manifolds, ordinary version] |
348
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|
1300 |
\label{ex:bord-cat} |
191
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|
1301 |
\rm |
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|
1302 |
\label{ex:bordism-category} |
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|
1303 |
For a $k$-ball $X$, $k<n$, define $\Bord^{n,d}(X)$ to be the set of all $(d{-}n{+}k)$-dimensional PL |
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|
1304 |
submanifolds $W$ of $X\times \Real^\infty$ such that $\bd W = W \cap \bd X \times \Real^\infty$. |
733
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|
1305 |
For an $n$-ball $X$ define $\Bord^{n,d}(X)$ to be homeomorphism classes (rel boundary) of such $d$-dimensional submanifolds; |
191
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|
1306 |
we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism |
196 | 1307 |
$W \to W'$ which restricts to the identity on the boundary. |
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|
1308 |
For $n=1$ we have the familiar bordism 1-category of $d$-manifolds. |
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|
1309 |
The case $n=d$ captures the $n$-categorical nature of bordisms. |
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|
1310 |
The case $n > 2d$ captures the full symmetric monoidal $n$-category structure. |
191
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|
1311 |
\end{example} |
833 | 1312 |
\begin{rem} |
737 | 1313 |
Working with the smooth bordism category would require careful attention to either collars, corners or halos. |
833 | 1314 |
\end{rem} |
191
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|
1315 |
|
196 | 1316 |
%\nn{the next example might be an unnecessary distraction. consider deleting it.} |
101 | 1317 |
|
196 | 1318 |
%\begin{example}[Variation on the above examples] |
1319 |
%We could allow $F$ to have boundary and specify boundary conditions on $X\times \bd F$, |
|
1320 |
%for example product boundary conditions or take the union over all boundary conditions. |
|
1321 |
%%\nn{maybe should not emphasize this case, since it's ``better" in some sense |
|
1322 |
%%to think of these guys as affording a representation |
|
1323 |
%%of the $n{+}1$-category associated to $\bd F$.} |
|
1324 |
%\end{example} |
|
101 | 1325 |
|
1326 |
||
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|
1327 |
%We have two main examples of $A_\infty$ $n$-categories, coming from maps to a target space and from the blob complex. |
101 | 1328 |
|
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|
1329 |
\begin{example}[Chains (or space) of maps to a space] |
191
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|
1330 |
\rm |
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|
1331 |
\label{ex:chains-of-maps-to-a-space} |
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|
1332 |
We can modify Example \ref{ex:maps-to-a-space} above to define the fundamental $A_\infty$ $n$-category $\pi^\infty_{\le n}(T)$ of a topological space $T$. |
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|
1333 |
For a $k$-ball $X$, with $k < n$, the set $\pi^\infty_{\leq n}(T)(X)$ is just $\Maps(X \to T)$. |
191
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|
1334 |
Define $\pi^\infty_{\leq n}(T)(X; c)$ for an $n$-ball $X$ and $c \in \pi^\infty_{\leq n}(T)(\bdy X)$ to be the chain complex |
418
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|
1335 |
\[ |
853
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|
1336 |
C_*(\Maps_c(X \to T)), |
418
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|
1337 |
\] |
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|
1338 |
where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
101 | 1339 |
and $C_*$ denotes singular chains. |
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|
1340 |
Alternatively, if we take the $n$-morphisms to be simply $\Maps_c(X \to T)$, |
888
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|
1341 |
we get an $A_\infty$ $n$-category enriched over spaces. |
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|
1342 |
\end{example} |
101 | 1343 |
|
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|
1344 |
See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to |
494
cb76847c439e
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|
1345 |
homotopy as the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
266
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|
1346 |
|
900
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|
1347 |
Instead of using the TQFT invariant $\cA$ as in Example \ref{ex:ncats-from-tqfts} above, we can turn an $n$-dimensional system of fields and local relations into an $A_\infty$ $n$-category using the blob complex. With a codimension $k$ fiber, we obtain an $A_\infty$ $k$-category: |
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|
1348 |
|
279 | 1349 |
\begin{example}[Blob complexes of balls (with a fiber)] |
1350 |
\rm |
|
1351 |
\label{ex:blob-complexes-of-balls} |
|
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|
1352 |
Fix an $n{-}k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$. |
888
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|
1353 |
We will define an $A_\infty$ $k$-category $\cC$. |
882 | 1354 |
When $X$ is an $m$-ball, with $m<k$, define $\cC(X) = \cE(X\times F)$. |
1355 |
When $X$ is a $k$-ball, |
|
279 | 1356 |
define $\cC(X; c) = \bc^\cE_*(X\times F; c)$ |
1357 |
where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
|
1358 |
\end{example} |
|
101 | 1359 |
|
445
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|
1360 |
This example will be used in Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product. |
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|
1361 |
Notice that with $F$ a point, the above example is a construction turning an ordinary |
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|
1362 |
$n$-category $\cC$ into an $A_\infty$ $n$-category. |
417
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|
1363 |
We think of this as providing a ``free resolution" |
888
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|
1364 |
of the ordinary $n$-category. |
475
07c18e2abd8f
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|
1365 |
%\nn{say something about cofibrant replacements?} |
340
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|
1366 |
In fact, there is also a trivial, but mostly uninteresting, way to do this: |
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|
1367 |
we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, |
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|
1368 |
and take $\CD{B}$ to act trivially. |
266
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|
1369 |
|
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|
1370 |
Beware that the ``free resolution" of the ordinary $n$-category $\pi_{\leq n}(T)$ |
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|
1371 |
is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. |
340
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|
1372 |
It's easy to see that with $n=0$, the corresponding system of fields is just |
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|
1373 |
linear combinations of connected components of $T$, and the local relations are trivial. |
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|
1374 |
There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
191
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|
1375 |
|
775 | 1376 |
\begin{example}[The bordism $n$-category of $d$-manifolds, $A_\infty$ version] |
309
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|
1377 |
\rm |
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|
1378 |
\label{ex:bordism-category-ainf} |
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changeset
|
1379 |
As in Example \ref{ex:bord-cat}, for $X$ a $k$-ball, $k<n$, we define $\Bord^{n,d}_\infty(X)$ |
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added 2nd parameter to the two bordism examples
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diff
changeset
|
1380 |
to be the set of all $(d{-}n{+}k)$-dimensional |
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|
1381 |
submanifolds $W$ of $X\times \Real^\infty$ such that $\bd W = W \cap \bd X \times \Real^\infty$. |
348
b2fab3bf491b
A-inf bordism cat example
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|
1382 |
For an $n$-ball $X$ with boundary condition $c$ |
733
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changeset
|
1383 |
define $\Bord^{n,d}_\infty(X; c)$ to be the space of all $d$-dimensional |
348
b2fab3bf491b
A-inf bordism cat example
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changeset
|
1384 |
submanifolds $W$ of $X\times \Real^\infty$ such that |
b2fab3bf491b
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diff
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|
1385 |
$W$ coincides with $c$ at $\bd X \times \Real^\infty$. |
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|
1386 |
(The topology on this space is induced by ambient isotopy rel boundary. |
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|
1387 |
This is homotopy equivalent to a disjoint union of copies $\mathrm{B}\!\Homeo(W')$, where |
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|
1388 |
$W'$ runs though representatives of homeomorphism types of such manifolds.) |
309
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|
1389 |
\end{example} |
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|
1390 |
|
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changeset
|
1391 |
|
346
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|
1392 |
|
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|
1393 |
Let $\cE\cB_n$ be the operad of smooth embeddings of $k$ (little) |
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|
1394 |
copies of the standard $n$-ball $B^n$ into another (big) copy of $B^n$. |
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|
1395 |
(We require that the interiors of the little balls be disjoint, but their |
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|
1396 |
boundaries are allowed to meet. |
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|
1397 |
Note in particular that the space for $k=1$ contains a copy of $\Diff(B^n)$, namely |
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|
1398 |
the embeddings of a ``little" ball with image all of the big ball $B^n$. |
475
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changeset
|
1399 |
(But note also that this inclusion is not |
781
0a9adf027f47
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parents:
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diff
changeset
|
1400 |
necessarily a homotopy equivalence.)) |
419
a571e37cc68d
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418
diff
changeset
|
1401 |
The operad $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad: |
a571e37cc68d
a few more ncat revisions
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418
diff
changeset
|
1402 |
by shrinking the little balls (precomposing them with dilations), |
346
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|
1403 |
we see that both operads are homotopic to the space of $k$ framed points |
401
a8b8ebcf07ac
Making notation in the product theorem more consistent.
Scott Morrison <scott@tqft.net>
parents:
400
diff
changeset
|
1404 |
in $B^n$. |
a8b8ebcf07ac
Making notation in the product theorem more consistent.
Scott Morrison <scott@tqft.net>
parents:
400
diff
changeset
|
1405 |
It is easy to see that $n$-fold loop spaces $\Omega^n(T)$ have |
346
90e0c5e7ae07
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|
1406 |
an action of $\cE\cB_n$. |
475
07c18e2abd8f
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diff
changeset
|
1407 |
%\nn{add citation for this operad if we can find one} |
346
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|
1408 |
|
309
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|
1409 |
\begin{example}[$E_n$ algebras] |
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|
1410 |
\rm |
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|
1411 |
\label{ex:e-n-alg} |
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|
1412 |
Let $A$ be an $\cE\cB_n$-algebra. |
346
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|
1413 |
Note that this implies a $\Diff(B^n)$ action on $A$, |
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|
1414 |
since $\cE\cB_n$ contains a copy of $\Diff(B^n)$. |
888
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changeset
|
1415 |
We will define a strict $A_\infty$ $n$-category $\cC^A$. |
869
c9df0c67af5d
minor intermediate commit, so that I can fetch
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diff
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|
1416 |
(We enrich in topological spaces, though this could easily be adapted to, say, chain complexes.) |
346
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|
1417 |
If $X$ is a ball of dimension $k<n$, define $\cC^A(X)$ to be a point. |
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|
1418 |
In other words, the $k$-morphisms are trivial for $k<n$. |
347
14643c4931bc
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|
1419 |
If $X$ is an $n$-ball, we define $\cC^A(X)$ via a colimit construction. |
14643c4931bc
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|
1420 |
(Plain colimit, not homotopy colimit.) |
14643c4931bc
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|
1421 |
Let $J$ be the category whose objects are embeddings of a disjoint union of copies of |
885
61541264d4b3
finishing most of the minor/typo issues from the referee
Scott Morrison <scott@tqft.net>
parents:
882
diff
changeset
|
1422 |
the standard ball $B^n$ into $X$, and whose morphisms are given by engulfing some of the |
347
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|
1423 |
embedded balls into a single larger embedded ball. |
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|
1424 |
To each object of $J$ we associate $A^{\times m}$ (where $m$ is the number of balls), and |
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|
1425 |
to each morphism of $J$ we associate a morphism coming from the $\cE\cB_n$ action on $A$. |
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|
1426 |
Alternatively and more simply, we could define $\cC^A(X)$ to be |
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|
1427 |
$\Diff(B^n\to X)\times A$ modulo the diagonal action of $\Diff(B^n)$. |
888
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|
1428 |
The remaining data for the $A_\infty$ $n$-category |
347
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|
1429 |
--- composition and $\Diff(X\to X')$ action --- |
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|
1430 |
also comes from the $\cE\cB_n$ action on $A$. |
528
96ec10a46ee1
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522
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|
1431 |
%\nn{should we spell this out?} |
346
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|
1432 |
|
888
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changeset
|
1433 |
Conversely, one can show that a disk-like strict $A_\infty$ $n$-category $\cC$, where the $k$-morphisms |
356
9bbe6eb6fb6c
remark about EB_n-algebras from n-cats
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diff
changeset
|
1434 |
$\cC(X)$ are trivial (single point) for $k<n$, gives rise to |
9bbe6eb6fb6c
remark about EB_n-algebras from n-cats
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diff
changeset
|
1435 |
an $\cE\cB_n$-algebra. |
528
96ec10a46ee1
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522
diff
changeset
|
1436 |
%\nn{The paper is already long; is it worth giving details here?} |
861
84bb5ab4c85c
unfinished edits to fam-o-homeo lemma and EB_n algebra example
Kevin Walker <kevin@canyon23.net>
parents:
859
diff
changeset
|
1437 |
% According to the referee, yes it is... |
84bb5ab4c85c
unfinished edits to fam-o-homeo lemma and EB_n algebra example
Kevin Walker <kevin@canyon23.net>
parents:
859
diff
changeset
|
1438 |
Let $A = \cC(B^n)$, where $B^n$ is the standard $n$-ball. |
869
c9df0c67af5d
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861
diff
changeset
|
1439 |
We must define maps |
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861
diff
changeset
|
1440 |
\[ |
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diff
changeset
|
1441 |
\cE\cB_n^k \times A \times \cdots \times A \to A , |
c9df0c67af5d
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parents:
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diff
changeset
|
1442 |
\] |
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changeset
|
1443 |
where $\cE\cB_n^k$ is the $k$-th space of the $\cE\cB_n$ operad. |
877 | 1444 |
Let $(b, a_1,\ldots,a_k)$ be a point of $\cE\cB_n^k \times A \times \cdots \times A \to A$. |
1445 |
The $i$-th embedding of $b$ together with $a_i$ determine an element of $\cC(B_i)$, |
|
1446 |
where $B_i$ denotes the $i$-th little ball. |
|
1447 |
Using composition of $n$-morphsims in $\cC$, and padding the spaces between the little balls with the |
|
1448 |
(essentially unique) identity $n$-morphism of $\cC$, we can construct a well-defined element |
|
1449 |
of $\cC(B^n) = A$. |
|
506 | 1450 |
|
1451 |
If we apply the homotopy colimit construction of the next subsection to this example, |
|
1452 |
we get an instance of Lurie's topological chiral homology construction. |
|
191
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diff
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|
1453 |
\end{example} |
95 | 1454 |
|
108 | 1455 |
|
310
ee7be19ee61a
converting sphere axiom to a proposition; still need to make similar changes in module axioms
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diff
changeset
|
1456 |
\subsection{From balls to manifolds} |
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changeset
|
1457 |
\label{ss:ncat_fields} \label{ss:ncat-coend} |
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changeset
|
1458 |
In this section we show how to extend an $n$-category $\cC$ as described above |
680
0591d017e698
plain n-cat -> ordinary n-cat
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parents:
679
diff
changeset
|
1459 |
(of either the ordinary or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$. |
552 | 1460 |
This extension is a certain colimit, and the arrow in the notation is intended as a reminder of this. |
1461 |
||
888
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changeset
|
1462 |
In the case of ordinary $n$-categories, this construction factors into a construction of a |
340
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|
1463 |
system of fields and local relations, followed by the usual TQFT definition of a |
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|
1464 |
vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}. |
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changeset
|
1465 |
For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. |
889 | 1466 |
Recall that we can take an ordinary $n$-category $\cC$ and pass to the ``free resolution", |
888
a0fd6e620926
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changeset
|
1467 |
an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls |
475
07c18e2abd8f
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changeset
|
1468 |
(recall Example \ref{ex:blob-complexes-of-balls} above). |
340
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|
1469 |
We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant |
888
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changeset
|
1470 |
for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the |
552 | 1471 |
same as the original blob complex for $M$ with coefficients in $\cC$. |
1472 |
||
818
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
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parents:
817
diff
changeset
|
1473 |
Recall that we've already anticipated this construction Subsection \ref{ss:n-cat-def}, |
552 | 1474 |
inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls, |
1475 |
so that we can state the boundary axiom for $\cC$ on $k+1$-balls. |
|
1476 |
||
1477 |
\medskip |
|
108 | 1478 |
|
781
0a9adf027f47
rewriting colimit def; there's still a little more to do
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diff
changeset
|
1479 |
We will first define the {\it decomposition poset} $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
888
a0fd6e620926
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diff
changeset
|
1480 |
An $n$-category $\cC$ provides a functor from this poset to the category of sets, |
419
a571e37cc68d
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|
1481 |
and we will define $\cl{\cC}(W)$ as a suitable colimit |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1482 |
(or homotopy colimit in the $A_\infty$ case) of this functor. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1483 |
We'll later give a more explicit description of this colimit. |
888
a0fd6e620926
Backed out changeset 7abe7642265e
Kevin Walker <kevin@canyon23.net>
parents:
865
diff
changeset
|
1484 |
In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain |
734
6fd9b377be3b
fix definition of refinement of ball decomp (intermediate manifolds are disj unions of balls)
Kevin Walker <kevin@canyon23.net>
parents:
733
diff
changeset
|
1485 |
complexes to $n$-balls with boundary data), |
6fd9b377be3b
fix definition of refinement of ball decomp (intermediate manifolds are disj unions of balls)
Kevin Walker <kevin@canyon23.net>
parents:
733
diff
changeset
|
1486 |
then the resulting colimit is also enriched, that is, the set associated to $W$ splits into |
6fd9b377be3b
fix definition of refinement of ball decomp (intermediate manifolds are disj unions of balls)
Kevin Walker <kevin@canyon23.net>
parents:
733
diff
changeset
|
1487 |
subsets according to boundary data, and each of these subsets has the appropriate structure |
6fd9b377be3b
fix definition of refinement of ball decomp (intermediate manifolds are disj unions of balls)
Kevin Walker <kevin@canyon23.net>
parents:
733
diff
changeset
|
1488 |
(e.g. a vector space or chain complex). |
108 | 1489 |
|
475
07c18e2abd8f
redefine "permissible decomp", and other changes to ntcat.tex; should be read
Kevin Walker <kevin@canyon23.net>
parents:
463
diff
changeset
|
1490 |
Recall (Definition \ref{defn:gluing-decomposition}) that a {\it ball decomposition} of $W$ is a |
07c18e2abd8f
redefine "permissible decomp", and other changes to ntcat.tex; should be read
Kevin Walker <kevin@canyon23.net>
parents:
463
diff
changeset
|
1491 |
sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls |
07c18e2abd8f
redefine "permissible decomp", and other changes to ntcat.tex; should be read
Kevin Walker <kevin@canyon23.net>
parents:
463
diff
changeset
|
1492 |
$\du_a X_a$. |
07c18e2abd8f
redefine "permissible decomp", and other changes to ntcat.tex; should be read
Kevin Walker <kevin@canyon23.net>
parents:
463
diff
changeset
|
1493 |
Abusing notation, we let $X_a$ denote both the ball (component of $M_0$) and |
07c18e2abd8f
redefine "permissible decomp", and other changes to ntcat.tex; should be read
Kevin Walker <kevin@canyon23.net>
parents:
463
diff
changeset
|
1494 |
its image in $W$ (which is not necessarily a ball --- parts of $\bd X_a$ may have been glued together). |
07c18e2abd8f
redefine "permissible decomp", and other changes to ntcat.tex; should be read
Kevin Walker <kevin@canyon23.net>
parents:
463
diff
changeset
|
1495 |
Define a {\it permissible decomposition} of $W$ to be a map |
108 | 1496 |
\[ |
475
07c18e2abd8f
redefine "permissible decomp", and other changes to ntcat.tex; should be read
Kevin Walker <kevin@canyon23.net>
parents:
463
diff
changeset
|
1497 |
\coprod_a X_a \to W, |
108 | 1498 |
\] |
475
07c18e2abd8f
redefine "permissible decomp", and other changes to ntcat.tex; should be read
Kevin Walker <kevin@canyon23.net>
parents:
463
diff
changeset
|
1499 |
which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$. |
818
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1500 |
We further require that $\du_a (X_a \cap \bd W) \to \bd W$ |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1501 |
can be completed to a (not necessarily ball) decomposition of $\bd W$. |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1502 |
(So, for example, in Example \ref{sin1x-example} if we take $W = B\cup C\cup D$ then $B\du C\du D \to W$ |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1503 |
is not allowed since $D\cap \bd W$ is not a submanifold.) |
475
07c18e2abd8f
redefine "permissible decomp", and other changes to ntcat.tex; should be read
Kevin Walker <kevin@canyon23.net>
parents:
463
diff
changeset
|
1504 |
Roughly, a permissible decomposition is like a ball decomposition where we don't care in which order the balls |
07c18e2abd8f
redefine "permissible decomp", and other changes to ntcat.tex; should be read
Kevin Walker <kevin@canyon23.net>
parents:
463
diff
changeset
|
1505 |
are glued up to yield $W$, so long as there is some (non-pathological) way to glue them. |
07c18e2abd8f
redefine "permissible decomp", and other changes to ntcat.tex; should be read
Kevin Walker <kevin@canyon23.net>
parents:
463
diff
changeset
|
1506 |
|
766
823999dd14fd
acknowledge the existence of manifolds without ball decompositions
Kevin Walker <kevin@canyon23.net>
parents:
758
diff
changeset
|
1507 |
(Every smooth or PL manifold has a ball decomposition, but certain topological manifolds (e.g.\ non-smoothable |
773
775b5ca42bed
make sure poset of decomps is a small category; added to to-do list
Kevin Walker <kevin@canyon23.net>
parents:
770
diff
changeset
|
1508 |
topological 4-manifolds) do not have ball decompositions. |
775b5ca42bed
make sure poset of decomps is a small category; added to to-do list
Kevin Walker <kevin@canyon23.net>
parents:
770
diff
changeset
|
1509 |
For such manifolds we have only the empty colimit.) |
775b5ca42bed
make sure poset of decomps is a small category; added to to-do list
Kevin Walker <kevin@canyon23.net>
parents:
770
diff
changeset
|
1510 |
|
775b5ca42bed
make sure poset of decomps is a small category; added to to-do list
Kevin Walker <kevin@canyon23.net>
parents:
770
diff
changeset
|
1511 |
We want the category (poset) of decompositions of $W$ to be small, so when we say decomposition we really |
775b5ca42bed
make sure poset of decomps is a small category; added to to-do list
Kevin Walker <kevin@canyon23.net>
parents:
770
diff
changeset
|
1512 |
mean isomorphism class of decomposition. |
775b5ca42bed
make sure poset of decomps is a small category; added to to-do list
Kevin Walker <kevin@canyon23.net>
parents:
770
diff
changeset
|
1513 |
Isomorphisms are defined in the obvious way: a collection of homeomorphisms $M_i\to M_i'$ which commute |
775b5ca42bed
make sure poset of decomps is a small category; added to to-do list
Kevin Walker <kevin@canyon23.net>
parents:
770
diff
changeset
|
1514 |
with the gluing maps $M_i\to M_{i+1}$ and $M'_i\to M'_{i+1}$. |
766
823999dd14fd
acknowledge the existence of manifolds without ball decompositions
Kevin Walker <kevin@canyon23.net>
parents:
758
diff
changeset
|
1515 |
|
479
cfad13b6b1e5
some modifications to blobdef
Scott Morrison <scott@tqft.net>
parents:
476
diff
changeset
|
1516 |
Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement |
475
07c18e2abd8f
redefine "permissible decomp", and other changes to ntcat.tex; should be read
Kevin Walker <kevin@canyon23.net>
parents:
463
diff
changeset
|
1517 |
of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$ |
734
6fd9b377be3b
fix definition of refinement of ball decomp (intermediate manifolds are disj unions of balls)
Kevin Walker <kevin@canyon23.net>
parents:
733
diff
changeset
|
1518 |
with $\du_b Y_b = M_i$ for some $i$, |
780
b76b4b79dbe1
starting to work on colimit stuff, but not much progress yet
Kevin Walker <kevin@canyon23.net>
parents:
775
diff
changeset
|
1519 |
and with $M_0, M_1, \ldots, M_i$ each being a disjoint union of balls. |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1520 |
|
419
a571e37cc68d
a few more ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
418
diff
changeset
|
1521 |
\begin{defn} |
479
cfad13b6b1e5
some modifications to blobdef
Scott Morrison <scott@tqft.net>
parents:
476
diff
changeset
|
1522 |
The poset $\cell(W)$ has objects the permissible decompositions of $W$, |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1523 |
and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
781
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1524 |
See Figure \ref{partofJfig}. |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1525 |
\end{defn} |
119 | 1526 |
|
774 | 1527 |
\begin{figure}[t] |
119 | 1528 |
\begin{equation*} |
222
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
218
diff
changeset
|
1529 |
\mathfig{.63}{ncat/zz2} |
119 | 1530 |
\end{equation*} |
329
eb03c4a92f98
various changes, mostly rewriting intros to sections for exposition
Scott Morrison <scott@tqft.net>
parents:
328
diff
changeset
|
1531 |
\caption{A small part of $\cell(W)$} |
119 | 1532 |
\label{partofJfig} |
1533 |
\end{figure} |
|
1534 |
||
888
a0fd6e620926
Backed out changeset 7abe7642265e
Kevin Walker <kevin@canyon23.net>
parents:
865
diff
changeset
|
1535 |
An $n$-category $\cC$ determines |
329
eb03c4a92f98
various changes, mostly rewriting intros to sections for exposition
Scott Morrison <scott@tqft.net>
parents:
328
diff
changeset
|
1536 |
a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
108 | 1537 |
(possibly with additional structure if $k=n$). |
781
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1538 |
Let $x = \{X_a\}$ be a permissible decomposition of $W$ (i.e.\ object of $\cD(W)$). |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1539 |
We will define $\psi_{\cC;W}(x)$ to be a certain subset of $\prod_a \cC(X_a)$. |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1540 |
Roughly speaking, $\psi_{\cC;W}(x)$ is the subset where the restriction maps from |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1541 |
$\cC(X_a)$ and $\cC(X_b)$ agree whenever some part of $\bd X_a$ is glued to some part of $\bd X_b$. |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1542 |
(Keep in mind that perhaps $a=b$.) |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1543 |
Since we allow decompositions in which the intersection of $X_a$ and $X_b$ might be messy |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1544 |
(see Example \ref{sin1x-example}), we must define $\psi_{\cC;W}(x)$ in a more roundabout way. |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1545 |
|
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1546 |
Inductively, we may assume that we have already defined the colimit $\cl\cC(M)$ for $k{-}1$-manifolds $M$. |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1547 |
(To start the induction, we define $\cl\cC(M)$, where $M = \du_a P_a$ is a 0-manifold and each $P_a$ is |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1548 |
a 0-ball, to be $\prod_a \cC(P_a)$.) |
783 | 1549 |
We also assume, inductively, that we have gluing and restriction maps for colimits of $k{-}1$-manifolds. |
784
bd9538de8248
more on colimits; still not done
Kevin Walker <kevin@canyon23.net>
parents:
783
diff
changeset
|
1550 |
Gluing and restriction maps for colimits of $k$-manifolds will be defined later in this subsection. |
781
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1551 |
|
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1552 |
Let $\du_a X_a = M_0\to\cdots\to M_m = W$ be a ball decomposition compatible with $x$. |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1553 |
Let $\bd M_i = N_i \cup Y_i \cup Y'_i$, where $Y_i$ and $Y'_i$ are glued together to produce $M_{i+1}$. |
833 | 1554 |
We will define $\psi_{\cC;W}(x)$ to be the subset of $\prod_a \cC(X_a)$ which satisfies a series of conditions |
781
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1555 |
related to the gluings $M_{i-1} \to M_i$, $1\le i \le m$. |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1556 |
By Axiom \ref{nca-boundary}, we have a map |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1557 |
\[ |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1558 |
\prod_a \cC(X_a) \to \cl\cC(\bd M_0) . |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1559 |
\] |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1560 |
The first condition is that the image of $\psi_{\cC;W}(x)$ in $\cl\cC(\bd M_0)$ is splittable |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1561 |
along $\bd Y_0$ and $\bd Y'_0$, and that the restrictions to $\cl\cC(Y_0)$ and $\cl\cC(Y'_0)$ agree |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1562 |
(with respect to the identification of $Y_0$ and $Y'_0$ provided by the gluing map). |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1563 |
|
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1564 |
On the subset of $\prod_a \cC(X_a)$ which satisfies the first condition above, we have a restriction |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1565 |
map to $\cl\cC(N_0)$ which we can compose with the gluing map |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1566 |
$\cl\cC(N_0) \to \cl\cC(\bd M_1)$. |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1567 |
The second condition is that the image of $\psi_{\cC;W}(x)$ in $\cl\cC(\bd M_1)$ is splittable |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1568 |
along $\bd Y_1$ and $\bd Y'_1$, and that the restrictions to $\cl\cC(Y_1)$ and $\cl\cC(Y'_1)$ agree |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1569 |
(with respect to the identification of $Y_1$ and $Y'_1$ provided by the gluing map). |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1570 |
The $i$-th condition is defined similarly. |
833 | 1571 |
Note that these conditions depend on the boundaries of elements of $\prod_a \cC(X_a)$. |
781
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1572 |
|
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1573 |
We define $\psi_{\cC;W}(x)$ to be the subset of $\prod_a \cC(X_a)$ which satisfies the |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1574 |
above conditions for all $i$ and also all |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1575 |
ball decompositions compatible with $x$. |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1576 |
(If $x$ is a nice, non-pathological cell decomposition, then it is easy to see that gluing |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1577 |
compatibility for one ball decomposition implies gluing compatibility for all other ball decompositions. |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1578 |
Rather than try to prove a similar result for arbitrary |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1579 |
permissible decompositions, we instead require compatibility with all ways of gluing up the decomposition.) |
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1580 |
|
784
bd9538de8248
more on colimits; still not done
Kevin Walker <kevin@canyon23.net>
parents:
783
diff
changeset
|
1581 |
If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ |
bd9538de8248
more on colimits; still not done
Kevin Walker <kevin@canyon23.net>
parents:
783
diff
changeset
|
1582 |
is given by the composition maps of $\cC$. |
bd9538de8248
more on colimits; still not done
Kevin Walker <kevin@canyon23.net>
parents:
783
diff
changeset
|
1583 |
This completes the definition of the functor $\psi_{\cC;W}$. |
108 | 1584 |
|
419
a571e37cc68d
a few more ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
418
diff
changeset
|
1585 |
If $k=n$ in the above definition and we are enriching in some auxiliary category, |
a571e37cc68d
a few more ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
418
diff
changeset
|
1586 |
we need to say a bit more. |
781
0a9adf027f47
rewriting colimit def; there's still a little more to do
Kevin Walker <kevin@canyon23.net>
parents:
780
diff
changeset
|
1587 |
We can rewrite the colimit as |
784
bd9538de8248
more on colimits; still not done
Kevin Walker <kevin@canyon23.net>
parents:
783
diff
changeset
|
1588 |
\[ % \begin{equation} \label{eq:psi-CC} |
419
a571e37cc68d
a few more ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
418
diff
changeset
|
1589 |
\psi_{\cC;W}(x) \deq \coprod_\beta \prod_a \cC(X_a; \beta) , |
784
bd9538de8248
more on colimits; still not done
Kevin Walker <kevin@canyon23.net>
parents:
783
diff
changeset
|
1590 |
\] % \end{equation} |
bd9538de8248
more on colimits; still not done
Kevin Walker <kevin@canyon23.net>
parents:
783
diff
changeset
|
1591 |
where $\beta$ runs through |
bd9538de8248
more on colimits; still not done
Kevin Walker <kevin@canyon23.net>
parents:
783
diff
changeset
|
1592 |
boundary conditions on $\du_a X_a$ which are compatible with gluing as specified above |
bd9538de8248
more on colimits; still not done
Kevin Walker <kevin@canyon23.net>
parents:
783
diff
changeset
|
1593 |
and $\cC(X_a; \beta)$ |
bd9538de8248
more on colimits; still not done
Kevin Walker <kevin@canyon23.net>
parents:
783
diff
changeset
|
1594 |
means the subset of $\cC(X_a)$ whose restriction to $\bd X_a$ agrees with $\beta$. |
419
a571e37cc68d
a few more ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
418
diff
changeset
|
1595 |
If we are enriching over $\cS$ and $k=n$, then $\cC(X_a; \beta)$ is an object in |
784
bd9538de8248
more on colimits; still not done
Kevin Walker <kevin@canyon23.net>
parents:
783
diff
changeset
|
1596 |
$\cS$ and the coproduct and product in the above expression should be replaced by the appropriate |
419
a571e37cc68d
a few more ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
418
diff
changeset
|
1597 |
operations in $\cS$ (e.g. direct sum and tensor product if $\cS$ is Vect). |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1598 |
|
494
cb76847c439e
many small fixes in ncat.tex
Scott Morrison <scott@tqft.net>
parents:
479
diff
changeset
|
1599 |
Finally, we construct $\cl{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$: |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
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190
diff
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|
1600 |
|
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1601 |
\begin{defn}[System of fields functor] |
415
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
1602 |
\label{def:colim-fields} |
888
a0fd6e620926
Backed out changeset 7abe7642265e
Kevin Walker <kevin@canyon23.net>
parents:
865
diff
changeset
|
1603 |
If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$. |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
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190
diff
changeset
|
1604 |
That is, for each decomposition $x$ there is a map |
402 | 1605 |
$\psi_{\cC;W}(x)\to \cl{\cC}(W)$, these maps are compatible with the refinement maps |
1606 |
above, and $\cl{\cC}(W)$ is universal with respect to these properties. |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1607 |
\end{defn} |
112 | 1608 |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1609 |
\begin{defn}[System of fields functor, $A_\infty$ case] |
888
a0fd6e620926
Backed out changeset 7abe7642265e
Kevin Walker <kevin@canyon23.net>
parents:
865
diff
changeset
|
1610 |
When $\cC$ is an $A_\infty$ $n$-category, $\cl{\cC}(W)$ for $W$ a $k$-manifold with $k < n$ |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1611 |
is defined as above, as the colimit of $\psi_{\cC;W}$. |
402 | 1612 |
When $W$ is an $n$-manifold, the chain complex $\cl{\cC}(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$. |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1613 |
\end{defn} |
8c2c330e87f2
working on ncats -- no new material, just improving text
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parents:
190
diff
changeset
|
1614 |
|
818
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1615 |
%We can specify boundary data $c \in \cl{\cC}(\bdy W)$, and define functors $\psi_{\cC;W,c}$ |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1616 |
%with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$. |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1617 |
|
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1618 |
\medskip |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1619 |
|
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1620 |
We must now define restriction maps $\bd : \cl{\cC}(W) \to \cl{\cC}(\bd W)$ and gluing maps. |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1621 |
|
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1622 |
Let $y\in \cl{\cC}(W)$. |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1623 |
Choose a representative of $y$ in the colimit: a permissible decomposition $\du_a X_a \to W$ and elements |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1624 |
$y_a \in \cC(X_a)$. |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1625 |
By assumption, $\du_a (X_a \cap \bd W) \to \bd W$ can be completed to a decomposition of $\bd W$. |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1626 |
Let $r(y_a) \in \cl\cC(X_a \cap \bd W)$ be the restriction. |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1627 |
Choose a representative of $r(y_a)$ in the colimit $\cl\cC(X_a \cap \bd W)$: a permissible decomposition |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1628 |
$\du_b Q_{ab} \to X_a \cap \bd W$ and elements $z_{ab} \in \cC(Q_{ab})$. |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1629 |
Then $\du_{ab} Q_{ab} \to \bd W$ is a permissible decomposition of $\bd W$ and $\{z_{ab}\}$ represents |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1630 |
an element of $\cl{\cC}(\bd W)$. Define $\bd y$ to be this element. |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1631 |
It is not hard to see that it is independent of the various choices involved. |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1632 |
|
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1633 |
Note that since we have already (inductively) defined gluing maps for colimits of $k{-}1$-manifolds, |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1634 |
we can also define restriction maps from $\cl{\cC}(W)\trans{}$ to $\cl{\cC}(Y)$ where $Y$ is a codimension 0 |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1635 |
submanifold of $\bd W$. |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1636 |
|
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1637 |
Next we define gluing maps for colimits of $k$-manifolds. |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1638 |
Let $W = W_1 \cup_Y W_2$. |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1639 |
Let $y_i \in \cl\cC(W_i)$ and assume that the restrictions of $y_1$ and $y_2$ to $\cl\cC(Y)$ agree. |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1640 |
We want to define $y_1\bullet y_2 \in \cl\cC(W)$. |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1641 |
Choose a permissible decomposition $\du_a X_{ia} \to W_i$ and elements |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1642 |
$y_{ia} \in \cC(X_{ia})$ representing $y_i$. |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1643 |
It might not be the case that $\du_{ia} X_{ia} \to W$ is a permissible decomposition of $W$, |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1644 |
since intersections of the pieces with $\bd W$ might not be well-behaved. |
896
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
1645 |
However, using the fact that $\bd y_i$ splits along $\bd Y$ and applying Axiom \ref{axiom:splittings}, |
818
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1646 |
we can choose the decomposition $\du_{a} X_{ia}$ so that its restriction to $\bd W_i$ is a refinement |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1647 |
of the splitting along $\bd Y$, and this implies that the combined decomposition $\du_{ia} X_{ia}$ |
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1648 |
is permissible. |
896
deeff619087e
Initial version of the new splitting axiom.
Kevin Walker <kevin@canyon23.net>
parents:
892
diff
changeset
|
1649 |
We can now define the gluing $y_1\bullet y_2$ in the obvious way, and a further application of Axiom \ref{axiom:splittings} |
833 | 1650 |
shows that this is independent of the choices of representatives of $y_i$. |
818
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1651 |
|
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1652 |
|
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1653 |
\medskip |
111 | 1654 |
|
422 | 1655 |
We now give more concrete descriptions of the above colimits. |
1656 |
||
1657 |
In the non-enriched case (e.g.\ $k<n$), where each $\cC(X_a; \beta)$ is just a set, |
|
1658 |
the colimit is |
|
1659 |
\[ |
|
494
cb76847c439e
many small fixes in ncat.tex
Scott Morrison <scott@tqft.net>
parents:
479
diff
changeset
|
1660 |
\cl{\cC}(W,c) = \left( \coprod_x \coprod_\beta \prod_a \cC(X_a; \beta) \right) \Bigg/ \sim , |
422 | 1661 |
\] |
818
fb9fc18d2a52
maybe done with colimit stuff; getting closer anyway
Kevin Walker <kevin@canyon23.net>
parents:
817
diff
changeset
|
1662 |
where $x$ runs through decompositions of $W$, and $\sim$ is the obvious equivalence relation |
422 | 1663 |
induced by refinement and gluing. |
833 | 1664 |
If $\cC$ is enriched over, for example, vector spaces and $W$ is an $n$-manifold, |
422 | 1665 |
we can take |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1666 |
\begin{equation*} |
494
cb76847c439e
many small fixes in ncat.tex
Scott Morrison <scott@tqft.net>
parents:
479
diff
changeset
|
1667 |
\cl{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) \Bigg/ K, |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1668 |
\end{equation*} |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1669 |
where $K$ is the vector space spanned by elements $a - g(a)$, with |
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1670 |
$a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
885
61541264d4b3
finishing most of the minor/typo issues from the referee
Scott Morrison <scott@tqft.net>
parents:
882
diff
changeset
|
1671 |
\to \psi_{\cC;W,c}(y)$ is the value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1672 |
|
225
32a76e8886d1
minor tweaks on small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
224
diff
changeset
|
1673 |
In the $A_\infty$ case, enriched over chain complexes, the concrete description of the homotopy colimit |
197 | 1674 |
is more involved. |
542
3baa4e4d395e
preparing for new def of morphisms of a-ing 1-cat modules
Kevin Walker <kevin@canyon23.net>
parents:
531
diff
changeset
|
1675 |
We will describe two different (but homotopy equivalent) versions of the homotopy colimit of $\psi_{\cC;W}$. |
3baa4e4d395e
preparing for new def of morphisms of a-ing 1-cat modules
Kevin Walker <kevin@canyon23.net>
parents:
531
diff
changeset
|
1676 |
The first is the usual one, which works for any indexing category. |
550 | 1677 |
The second construction, which we call the {\it local} homotopy colimit, |
542
3baa4e4d395e
preparing for new def of morphisms of a-ing 1-cat modules
Kevin Walker <kevin@canyon23.net>
parents:
531
diff
changeset
|
1678 |
is more closely related to the blob complex |
3baa4e4d395e
preparing for new def of morphisms of a-ing 1-cat modules
Kevin Walker <kevin@canyon23.net>
parents:
531
diff
changeset
|
1679 |
construction of \S \ref{sec:blob-definition} and takes advantage of local (gluing) properties |
3baa4e4d395e
preparing for new def of morphisms of a-ing 1-cat modules
Kevin Walker <kevin@canyon23.net>
parents:
531
diff
changeset
|
1680 |
of the indexing category $\cell(W)$. |
3baa4e4d395e
preparing for new def of morphisms of a-ing 1-cat modules
Kevin Walker <kevin@canyon23.net>
parents:
531
diff
changeset
|
1681 |
|
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1682 |
Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$. |
329
eb03c4a92f98
various changes, mostly rewriting intros to sections for exposition
Scott Morrison <scott@tqft.net>
parents:
328
diff
changeset
|
1683 |
Such sequences (for all $m$) form a simplicial set in $\cell(W)$. |
402 | 1684 |
Define $\cl{\cC}(W)$ as a vector space via |
112 | 1685 |
\[ |
402 | 1686 |
\cl{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
112 | 1687 |
\] |
494
cb76847c439e
many small fixes in ncat.tex
Scott Morrison <scott@tqft.net>
parents:
479
diff
changeset
|
1688 |
where the sum is over all $m$ and all $m$-sequences $(x_i)$, and each summand is degree shifted by $m$. |
463 | 1689 |
Elements of a summand indexed by an $m$-sequence will be call $m$-simplices. |
402 | 1690 |
We endow $\cl{\cC}(W)$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ |
112 | 1691 |
summands plus another term using the differential of the simplicial set of $m$-sequences. |
1692 |
More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
|
402 | 1693 |
summand of $\cl{\cC}(W)$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
112 | 1694 |
\[ |
191
8c2c330e87f2
working on ncats -- no new material, just improving text
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
190
diff
changeset
|
1695 |
\bd (a, \bar{x}) = (\bd a, \bar{x}) + (-1)^{\deg{a}} (g(a), d_0(\bar{x})) + (-1)^{\deg{a}} \sum_{j=1}^k (-1)^{j} (a, d_j(\bar{x})) , |
112 | 1696 |
\] |
1697 |
where $d_j(\bar{x}) = (x_0,\dots,x_{j-1},x_{j+1},\dots,x_k)$ and $g: \psi_\cC(x_0)\to \psi_\cC(x_1)$ |
|
198 | 1698 |
is the usual gluing map coming from the antirefinement $x_0 \le x_1$. |
422 | 1699 |
%\nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which |
1700 |
%combine only two balls at a time; for $n=1$ this version will lead to usual definition |
|
1701 |
%of $A_\infty$ category} |
|
108 | 1702 |
|
113 | 1703 |
We can think of this construction as starting with a disjoint copy of a complex for each |
461
c04bb911d636
changing simplex terminology for hocolimit (no more "degree")
Kevin Walker <kevin@canyon23.net>
parents:
456
diff
changeset
|
1704 |
permissible decomposition (the 0-simplices). |
113 | 1705 |
Then we glue these together with mapping cylinders coming from gluing maps |
461
c04bb911d636
changing simplex terminology for hocolimit (no more "degree")
Kevin Walker <kevin@canyon23.net>
parents:
456
diff
changeset
|
1706 |
(the 1-simplices). |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1707 |
Then we kill the extra homology we just introduced with mapping |
461
c04bb911d636
changing simplex terminology for hocolimit (no more "degree")
Kevin Walker <kevin@canyon23.net>
parents:
456
diff
changeset
|
1708 |
cylinders between the mapping cylinders (the 2-simplices), and so on. |
113 | 1709 |
|
542
3baa4e4d395e
preparing for new def of morphisms of a-ing 1-cat modules
Kevin Walker <kevin@canyon23.net>
parents:
531
diff
changeset
|
1710 |
Next we describe the local homotopy colimit. |
3baa4e4d395e
preparing for new def of morphisms of a-ing 1-cat modules
Kevin Walker <kevin@canyon23.net>
parents:
531
diff
changeset
|
1711 |
This is similar to the usual homotopy colimit, but using |
3baa4e4d395e
preparing for new def of morphisms of a-ing 1-cat modules
Kevin Walker <kevin@canyon23.net>
parents:
531
diff
changeset
|
1712 |
a cone-product set (Remark \ref{blobsset-remark}) in place of a simplicial set. |
3baa4e4d395e
preparing for new def of morphisms of a-ing 1-cat modules
Kevin Walker <kevin@canyon23.net>
parents:
531
diff
changeset
|
1713 |
The cone-product $m$-polyhedra for the set are pairs $(x, E)$, where $x$ is a decomposition of $W$ |
3baa4e4d395e
preparing for new def of morphisms of a-ing 1-cat modules
Kevin Walker <kevin@canyon23.net>
parents:
531
diff
changeset
|
1714 |
and $E$ is an $m$-blob diagram such that each blob is a union of balls of $x$. |
3baa4e4d395e
preparing for new def of morphisms of a-ing 1-cat modules
Kevin Walker <kevin@canyon23.net>
parents:
531
diff
changeset
|
1715 |
(Recall that this means that the interiors of |
3baa4e4d395e
preparing for new def of morphisms of a-ing 1-cat modules
Kevin Walker <kevin@canyon23.net>
parents:
531
diff
changeset
|
1716 |
each pair of blobs (i.e.\ balls) of $E$ are either disjoint or nested.) |
3baa4e4d395e
preparing for new def of morphisms of a-ing 1-cat modules
Kevin Walker <kevin@canyon23.net>
parents:
531
diff
changeset
|
1717 |
To each $(x, E)$ we associate the chain complex $\psi_{\cC;W}(x)$, shifted in degree by $m$. |
3baa4e4d395e
preparing for new def of morphisms of a-ing 1-cat modules
Kevin Walker <kevin@canyon23.net>
parents:
531
diff
changeset
|
1718 |
The boundary has a term for omitting each blob of $E$. |
3baa4e4d395e
preparing for new def of morphisms of a-ing 1-cat modules
Kevin Walker <kevin@canyon23.net>
parents:
531
diff
changeset
|
1719 |
If we omit an innermost blob then we replace $x$ by the formal difference $x - \gl(x)$, where |
3baa4e4d395e
preparing for new def of morphisms of a-ing 1-cat modules
Kevin Walker <kevin@canyon23.net>
parents:
531
diff
changeset
|
1720 |
$\gl(x)$ is obtained from $x$ by gluing together the balls of $x$ contained in the blob we are omitting. |
3baa4e4d395e
preparing for new def of morphisms of a-ing 1-cat modules
Kevin Walker <kevin@canyon23.net>
parents:
531
diff
changeset
|
1721 |
The gluing maps of $\cC$ give us a maps from $\psi_{\cC;W}(x)$ to $\psi_{\cC;W}(\gl(x))$. |
3baa4e4d395e
preparing for new def of morphisms of a-ing 1-cat modules
Kevin Walker <kevin@canyon23.net>
parents:
531
diff
changeset
|
1722 |
|
3baa4e4d395e
preparing for new def of morphisms of a-ing 1-cat modules
Kevin Walker <kevin@canyon23.net>
parents:
531
diff
changeset
|
1723 |
One can show that the usual hocolimit and the local hocolimit are homotopy equivalent using an |
3baa4e4d395e
preparing for new def of morphisms of a-ing 1-cat modules
Kevin Walker <kevin@canyon23.net>
parents:
531
diff
changeset
|
1724 |
Eilenberg-Zilber type subdivision argument. |
3baa4e4d395e
preparing for new def of morphisms of a-ing 1-cat modules
Kevin Walker <kevin@canyon23.net>
parents:
531
diff
changeset
|
1725 |
|
3baa4e4d395e
preparing for new def of morphisms of a-ing 1-cat modules
Kevin Walker <kevin@canyon23.net>
parents:
531
diff
changeset
|
1726 |
\medskip |
3baa4e4d395e
preparing for new def of morphisms of a-ing 1-cat modules
Kevin Walker <kevin@canyon23.net>
parents:
531
diff
changeset
|
1727 |
|
552 | 1728 |
$\cl{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
1729 |
Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}. |
|
108 | 1730 |
|
415
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
1731 |
\begin{lem} |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
1732 |
\label{lem:colim-injective} |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
1733 |
Let $W$ be a manifold of dimension less than $n$. Then for each |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
1734 |
decomposition $x$ of $W$ the natural map $\psi_{\cC;W}(x)\to \cl{\cC}(W)$ is injective. |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
1735 |
\end{lem} |
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
1736 |
\begin{proof} |
531
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1737 |
$\cl{\cC}(W)$ is a colimit of a diagram of sets, and each of the arrows in the diagram is |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1738 |
injective. |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1739 |
Concretely, the colimit is the disjoint union of the sets (one for each decomposition of $W$), |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1740 |
modulo the relation which identifies the domain of each of the injective maps |
773
775b5ca42bed
make sure poset of decomps is a small category; added to to-do list
Kevin Walker <kevin@canyon23.net>
parents:
770
diff
changeset
|
1741 |
with its image. |
531
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1742 |
|
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1743 |
To save ink and electrons we will simplify notation and write $\psi(x)$ for $\psi_{\cC;W}(x)$. |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1744 |
|
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1745 |
Suppose $a, \hat{a}\in \psi(x)$ have the same image in $\cl{\cC}(W)$ but $a\ne \hat{a}$. |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1746 |
Then there exist |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1747 |
\begin{itemize} |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1748 |
\item decompositions $x = x_0, x_1, \ldots , x_{k-1}, x_k = x$ and $v_1,\ldots, v_k$ of $W$; |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1749 |
\item anti-refinements $v_i\to x_i$ and $v_i\to x_{i-1}$; and |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1750 |
\item elements $a_i\in \psi(x_i)$ and $b_i\in \psi(v_i)$, with $a_0 = a$ and $a_k = \hat{a}$, |
809 | 1751 |
such that $b_i$ and $b_{i+1}$ both map to (glue up to) $a_i$. |
531
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1752 |
\end{itemize} |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1753 |
In other words, we have a zig-zag of equivalences starting at $a$ and ending at $\hat{a}$. |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1754 |
The idea of the proof is to produce a similar zig-zag where everything antirefines to the same |
535
07b79f81c956
numbering axioms and module axioms as 7.x
Scott Morrison <scott@tqft.net>
parents:
531
diff
changeset
|
1755 |
disjoint union of balls, and then invoke Axiom \ref{nca-assoc} which ensures associativity. |
531
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1756 |
|
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1757 |
Let $z$ be a decomposition of $W$ which is in general position with respect to all of the |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1758 |
$x_i$'s and $v_i$'s. |
885
61541264d4b3
finishing most of the minor/typo issues from the referee
Scott Morrison <scott@tqft.net>
parents:
882
diff
changeset
|
1759 |
There exist decompositions $x'_i$ and $v'_i$ (for all $i$) such that |
531
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1760 |
\begin{itemize} |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1761 |
\item $x'_i$ antirefines to $x_i$ and $z$; |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1762 |
\item $v'_i$ antirefines to $x'_i$, $x'_{i-1}$ and $v_i$; |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1763 |
\item $b_i$ is the image of some $b'_i\in \psi(v'_i)$; and |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1764 |
\item $a_i$ is the image of some $a'_i\in \psi(x'_i)$, which in turn is the image |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1765 |
of $b'_i$ and $b'_{i+1}$. |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1766 |
\end{itemize} |
914
db365e67adf6
finished with plitting axiom stuff (except that now the module definition needs to be updated)
Kevin Walker <kevin@canyon23.net>
parents:
913
diff
changeset
|
1767 |
(This is possible by Axiom \ref{axiom:splittings}.) |
531
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1768 |
Now consider the diagrams |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1769 |
\[ \xymatrix{ |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1770 |
& \psi(x'_{i-1}) \ar[rd] & \\ |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1771 |
\psi(v'_i) \ar[ru] \ar[rd] & & \psi(z) \\ |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1772 |
& \psi(x'_i) \ar[ru] & |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1773 |
} \] |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1774 |
The associativity axiom applied to this diagram implies that $a'_{i-1}$ and $a'_i$ |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1775 |
map to the same element $c\in \psi(z)$. |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1776 |
Therefore $a'_0$ and $a'_k$ both map to $c$. |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1777 |
But $a'_0$ and $a'_k$ are both elements of $\psi(x'_0)$ (because $x'_k = x'_0$). |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1778 |
So by the injectivity clause of the composition axiom, we must have that $a'_0 = a'_k$. |
da9bf150bf3d
proof of injectivity/colimit lemma
Kevin Walker <kevin@canyon23.net>
parents:
530
diff
changeset
|
1779 |
But this implies that $a = a_0 = a_k = \hat{a}$, contrary to our assumption that $a\ne \hat{a}$. |
415
8dedd2914d10
starting to revise ncat section
Kevin Walker <kevin@canyon23.net>
parents:
411
diff
changeset
|
1780 |
\end{proof} |
402 | 1781 |
|
552 | 1782 |
%\nn{need to finish explaining why we have a system of fields; |
1783 |
%define $k$-cat $\cC(\cdot\times W)$} |
|
108 | 1784 |
|
1785 |
\subsection{Modules} |
|
867
d7130746cfad
adding some forward references about extended TQFTs, per referee
Scott Morrison <scott@tqft.net>
parents:
866
diff
changeset
|
1786 |
\label{sec:modules} |
888
a0fd6e620926
Backed out changeset 7abe7642265e
Kevin Walker <kevin@canyon23.net>
parents:
865
diff
changeset
|
1787 |
Next we define ordinary and $A_\infty$ $n$-category modules. |
a0fd6e620926
Backed out changeset 7abe7642265e
Kevin Walker <kevin@canyon23.net>
parents:
865
diff
changeset
|
1788 |
The definition will be very similar to that of $n$-categories, |
199 | 1789 |
but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
198 | 1790 |
|
104 | 1791 |
Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
102 | 1792 |
in the context of an $m{+}1$-dimensional TQFT. |
888
a0fd6e620926
Backed out changeset 7abe7642265e
Kevin Walker <kevin@canyon23.net>
parents:
865
diff
changeset
|
1793 |
Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
102 | 1794 |
This will be explained in more detail as we present the axioms. |
1795 |
||
888
a0fd6e620926
Backed out changeset 7abe7642265e
Kevin Walker <kevin@canyon23.net>
parents:
865
diff
changeset
|
1796 |
Throughout, we fix an $n$-category $\cC$. |
685
8efbd2730ef9
"topological n-cat" --> either "disk-like n-cat" or "ordinary n-cat" (when contrasted with A-inf n-cat)
Kevin Walker <kevin@canyon23.net>
parents:
683
diff
changeset
|
1797 |
For all but one axiom, it doesn't matter whether $\cC$ is an ordinary $n$-category or an $A_\infty$ $n$-category. |
494
cb76847c439e
many small fixes in ncat.tex
Scott Morrison <scott@tqft.net>
parents:
479
diff
changeset
|
1798 |
We state the final axiom, regarding actions of homeomorphisms, differently in the two cases. |
102 | 1799 |
|
1800 |
Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair |
|
222
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
218
diff
changeset
|
1801 |
$$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$ |
102 | 1802 |
We call $B$ the ball and $N$ the marking. |
1803 |
A homeomorphism between marked $k$-balls is a homeomorphism of balls which |
|
1804 |
restricts to a homeomorphism of markings. |
|
1805 |
||
546
689ef4edbdd7
new def of mophisms between modules
Kevin Walker <kevin@canyon23.net>
parents:
543
diff
changeset
|
1806 |
\begin{module-axiom}[Module morphisms] \label{module-axiom-funct} |
904
fab3d057beeb
marked balls start at k=1, not k=0
Kevin Walker <kevin@canyon23.net>
parents:
903
diff
changeset
|
1807 |
{For each $1 \le k \le n$, we have a functor $\cM_k$ from |
102 | 1808 |
the category of marked $k$-balls and |
1809 |
homeomorphisms to the category of sets and bijections.} |
|
336
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replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
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diff
changeset
|
1810 |
\end{module-axiom} |
102 | 1811 |
|
1812 |
(As with $n$-categories, we will usually omit the subscript $k$.) |
|
1813 |
||
423 | 1814 |
For example, let $\cD$ be the TQFT which assigns to a $k$-manifold $N$ the set |
1815 |
of maps from $N$ to $T$ (for $k\le m$), modulo homotopy (and possibly linearized) if $k=m$. |
|
104 | 1816 |
Let $W$ be an $(m{-}n{+}1)$-dimensional manifold with boundary. |
1817 |
Let $\cC$ be the $n$-category with $\cC(X) \deq \cD(X\times \bd W)$. |
|
423 | 1818 |
Let $\cM(B, N) \deq \cD((B\times \bd W)\cup (N\times W))$ |
1819 |
(see Example \ref{ex:maps-with-fiber}). |
|
104 | 1820 |
(The union is along $N\times \bd W$.) |
423 | 1821 |
%(If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be |
1822 |
%the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.) |
|
102 | 1823 |
|
774 | 1824 |
\begin{figure}[t] |
494
cb76847c439e
many small fixes in ncat.tex
Scott Morrison <scott@tqft.net>
parents:
479
diff
changeset
|
1825 |
$$\mathfig{.55}{ncat/boundary-collar}$$ |
182 | 1826 |
\caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure} |
1827 |
||
103 | 1828 |
Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
778
760cc71a0424
add remarks to address the bizarre and inexplicable confusion about "hemisphere"
Kevin Walker <kevin@canyon23.net>
parents:
775
diff
changeset
|
1829 |
Call such a thing a {\it marked $k{-}1$-hemisphere}. |
760cc71a0424
add remarks to address the bizarre and inexplicable confusion about "hemisphere"
Kevin Walker <kevin@canyon23.net>
parents:
775
diff
changeset
|
1830 |
(A marked $k{-}1$-hemisphere is, of course, just a $k{-}1$-ball with its entire boundary marked. |
760cc71a0424
add remarks to address the bizarre and inexplicable confusion about "hemisphere"
Kevin Walker <kevin@canyon23.net>
parents:
775
diff
changeset
|
1831 |
We call it a hemisphere instead of a ball because it plays a role analogous |
760cc71a0424
add remarks to address the bizarre and inexplicable confusion about "hemisphere"
Kevin Walker <kevin@canyon23.net>
parents:
775
diff
changeset
|
1832 |
to the $k{-}1$-spheres in the $n$-category definition.) |
102 | 1833 |
|
336
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
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335
diff
changeset
|
1834 |
\begin{lem} |
7a5a73ec8961
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335
diff
changeset
|
1835 |
\label{lem:hemispheres} |
424
6ebf92d2ccef
ncat.tex mostly module stuff
Kevin Walker <kevin@canyon23.net>
parents:
423
diff
changeset
|
1836 |
{For each $0 \le k \le n-1$, we have a functor $\cl\cM_k$ from |
104 | 1837 |
the category of marked $k$-hemispheres and |
102 | 1838 |
homeomorphisms to the category of sets and bijections.} |
336
7a5a73ec8961
replacing axioms with lemmas in the module section; still out of sync with the ncat axioms
Scott Morrison <scott@tqft.net>
parents:
335
diff
changeset
|
1839 |
\end{lem} |
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1840 |
The proof is exactly analogous to that of Lemma \ref{lem:spheres}, and we omit the details. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1841 |
We use the same type of colimit construction. |
102 | 1842 |
|
424
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1843 |
In our example, $\cl\cM(H) = \cD(H\times\bd W \cup \bd H\times W)$. |
104 | 1844 |
|
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1845 |
\begin{module-axiom}[Module boundaries] |
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1846 |
{For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cl\cM(\bd M)$. |
102 | 1847 |
These maps, for various $M$, comprise a natural transformation of functors.} |
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1848 |
\end{module-axiom} |
102 | 1849 |
|
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1850 |
Given $c\in\cl\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
102 | 1851 |
|
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1852 |
If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
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1853 |
then for each marked $n$-ball $M=(B,N)$ and $c\in \cC(\bd B \setminus N)$, the set $\cM(M; c)$ should be an object in that category. |
102 | 1854 |
|
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1855 |
\begin{lem}[Boundary from domain and range] |
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|
1856 |
\label{lem:module-boundary} |
423 | 1857 |
{Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k{-}1$-hemisphere ($1\le k\le n$), |
1858 |
$M_i$ is a marked $k{-}1$-ball, and $E = M_1\cap M_2$ is a marked $k{-}2$-hemisphere. |
|
104 | 1859 |
Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the |
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1860 |
two maps $\bd: \cM(M_i)\to \cl\cM(E)$. |
423 | 1861 |
Then we have an injective map |
102 | 1862 |
\[ |
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1863 |
\gl_E : \cM(M_1) \times_{\cl\cM(E)} \cM(M_2) \hookrightarrow \cl\cM(H) |
102 | 1864 |
\] |
1865 |
which is natural with respect to the actions of homeomorphisms.} |
|
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1866 |
\end{lem} |
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1867 |
Again, this is in exact analogy with Lemma \ref{lem:domain-and-range}, and illustrated in Figure \ref{fig:module-boundary}. |
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|
1868 |
\begin{figure}[t] |
880 | 1869 |
\tikzset{marked/.style={line width=3pt}} |
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|
1870 |
|
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1871 |
\begin{equation*} |
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1872 |
\begin{tikzpicture}[baseline=0] |
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|
1873 |
\coordinate (a) at (0,1); |
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|
1874 |
\coordinate (b) at (4,1); |
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|
1875 |
\draw[marked] (a) arc (180:0:2); |
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|
1876 |
\draw (b) -- (a); |
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|
1877 |
\node at (2,2) {$M_1$}; |
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|
1878 |
|
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|
1879 |
\draw (0,0) node[fill, circle] {} -- (4,0) node[fill,circle] {}; |
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|
1880 |
\node at (-0.6,0) {$E$}; |
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|
1881 |
|
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|
1882 |
\draw[marked] (0,-1) arc(-180:0:2); |
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|
1883 |
\draw (4,-1) -- (0,-1); |
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|
1884 |
\node at (2,-2) {$M_2$}; |
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|
1885 |
\end{tikzpicture} |
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|
1886 |
\qquad \qquad \qquad |
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|
1887 |
\begin{tikzpicture}[baseline=0] |
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|
1888 |
\draw[marked] (0,0) node {$H$} circle (2); |
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|
1889 |
\end{tikzpicture} |
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|
1890 |
\end{equation*}\caption{The marked hemispheres and marked balls from Lemma \ref{lem:module-boundary}.} |
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|
1891 |
\label{fig:module-boundary} |
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|
1892 |
\end{figure} |
102 | 1893 |
|
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|
1894 |
Let $\cl\cM(H)\trans E$ denote the image of $\gl_E$. |
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|
1895 |
We will refer to elements of $\cl\cM(H)\trans E$ as ``splittable along $E$" or ``transverse to $E$". |
110 | 1896 |
|
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|
1897 |
\noop{ %%%%%%% |
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|
1898 |
\begin{lem}[Module to category restrictions] |
103 | 1899 |
{For each marked $k$-hemisphere $H$ there is a restriction map |
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|
1900 |
$\cl\cM(H)\to \cC(H)$. |
103 | 1901 |
($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) |
1902 |
These maps comprise a natural transformation of functors.} |
|
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1903 |
\end{lem} |
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|
1904 |
} %%%%%%% end \noop |
102 | 1905 |
|
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|
1906 |
It follows from the definition of the colimit $\cl\cM(H)$ that |
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|
1907 |
given any (unmarked) $k{-}1$-ball $Y$ in the interior of $H$ there is a restriction map |
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|
1908 |
from a subset $\cl\cM(H)_{\trans{\bdy Y}}$ of $\cl\cM(H)$ to $\cC(Y)$. |
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|
1909 |
Combining this with the boundary map $\cM(B,N) \to \cl\cM(\bd(B,N))$, we also have a restriction |
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|
1910 |
map from a subset $\cM(B,N)_{\trans{\bdy Y}}$ of $\cM(B,N)$ to $\cC(Y)$ whenever $Y$ is in the interior of $\bd B \setmin N$. |
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|
1911 |
This fact will be used below. |
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|
1912 |
|
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|
1913 |
\noop{ %%%% |
103 | 1914 |
Note that combining the various boundary and restriction maps above |
110 | 1915 |
(for both modules and $n$-categories) |
103 | 1916 |
we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$ |
1917 |
a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$. |
|
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|
1918 |
This subset $\cM(B,N)\trans{\bdy Y}$ is the subset of morphisms which are appropriately splittable (transverse to the |
110 | 1919 |
cutting submanifolds). |
103 | 1920 |
This fact will be used below. |
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|
1921 |
} %%%%% end \noop |
102 | 1922 |
|
104 | 1923 |
In our example, the various restriction and gluing maps above come from |
1924 |
restricting and gluing maps into $T$. |
|
1925 |
||
1926 |
We require two sorts of composition (gluing) for modules, corresponding to two ways |
|
103 | 1927 |
of splitting a marked $k$-ball into two (marked or plain) $k$-balls. |
119 | 1928 |
(See Figure \ref{zzz3}.) |
103 | 1929 |
|
774 | 1930 |
\begin{figure}[t] |
119 | 1931 |
\begin{equation*} |
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|
1932 |
\mathfig{.4}{ncat/zz3} |
119 | 1933 |
\end{equation*} |
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|
1934 |
\caption{Module composition (top); $n$-category action (bottom).} |
119 | 1935 |
\label{zzz3} |
1936 |
\end{figure} |
|
1937 |
||
1938 |
First, we can compose two module morphisms to get another module morphism. |
|
103 | 1939 |
|
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|
1940 |
\begin{module-axiom}[Module composition] |
222
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1941 |
{Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls (with $0\le k\le n$) |
103 | 1942 |
and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball. |
1943 |
Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere. |
|
1944 |
Note that each of $M$, $M_1$ and $M_2$ has its boundary split into two marked $k{-}1$-balls by $E$. |
|
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|
1945 |
We have restriction (domain or range) maps $\cM(M_i)\trans E \to \cM(Y)$. |
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|
1946 |
Let $\cM(M_1) \trans E \times_{\cM(Y)} \cM(M_2) \trans E$ denote the fibered product of these two maps. |
103 | 1947 |
Then (axiom) we have a map |
1948 |
\[ |
|
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|
1949 |
\gl_Y : \cM(M_1) \trans E \times_{\cM(Y)} \cM(M_2) \trans E \to \cM(M) \trans E |
103 | 1950 |
\] |
1951 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
1952 |
to the intersection of the boundaries of $M$ and $M_i$. |
|
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|
1953 |
If $k < n$ we require that $\gl_Y$ is injective.} |
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|
1954 |
\end{module-axiom} |
119 | 1955 |
|
1956 |
||
103 | 1957 |
Second, we can compose an $n$-category morphism with a module morphism to get another |
1958 |
module morphism. |
|
1959 |
We'll call this the action map to distinguish it from the other kind of composition. |
|
1960 |
||
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|
1961 |
\begin{module-axiom}[$n$-category action] |
103 | 1962 |
{Let $M = X \cup_Y M'$, where $M$ and $M'$ are marked $k$-balls ($0\le k\le n$), |
1963 |
$X$ is a plain $k$-ball, |
|
1964 |
and $Y = X\cap M'$ is a $k{-}1$-ball. |
|
1965 |
Let $E = \bd Y$, which is a $k{-}2$-sphere. |
|
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|
1966 |
We have restriction maps $\cM(M') \trans E \to \cC(Y)$ and $\cC(X) \trans E\to \cC(Y)$. |
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|
1967 |
Let $\cC(X)\trans E \times_{\cC(Y)} \cM(M') \trans E$ denote the fibered product of these two maps. |
103 | 1968 |
Then (axiom) we have a map |
1969 |
\[ |
|
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|
1970 |
\gl_Y :\cC(X)\trans E \times_{\cC(Y)} \cM(M')\trans E \to \cM(M) \trans E |
103 | 1971 |
\] |
1972 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
1973 |
to the intersection of the boundaries of $X$ and $M'$. |
|
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|
1974 |
If $k < n$ we require that $\gl_Y$ is injective.} |
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|
1975 |
\end{module-axiom} |
103 | 1976 |
|
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|
1977 |
\begin{module-axiom}[Strict associativity] |
423 | 1978 |
The composition and action maps above are strictly associative. |
475
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|
1979 |
Given any decomposition of a large marked ball into smaller marked and unmarked balls |
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|
1980 |
any sequence of pairwise gluings yields (via composition and action maps) the same result. |
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|
1981 |
\end{module-axiom} |
103 | 1982 |
|
110 | 1983 |
Note that the above associativity axiom applies to mixtures of module composition, |
1984 |
action maps and $n$-category composition. |
|
119 | 1985 |
See Figure \ref{zzz1b}. |
1986 |
||
774 | 1987 |
\begin{figure}[t] |
119 | 1988 |
\begin{equation*} |
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|
1989 |
\mathfig{0.49}{ncat/zz0} \mathfig{0.49}{ncat/zz1} |
119 | 1990 |
\end{equation*} |
1991 |
\caption{Two examples of mixed associativity} |
|
1992 |
\label{zzz1b} |
|
1993 |
\end{figure} |
|
1994 |
||
110 | 1995 |
|
1996 |
The above three axioms are equivalent to the following axiom, |
|
103 | 1997 |
which we state in slightly vague form. |
1998 |
||
1999 |
\xxpar{Module multi-composition:} |
|
494
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|
2000 |
{Given any splitting |
103 | 2001 |
\[ |
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|
2002 |
X_1 \sqcup\cdots\sqcup X_p \sqcup M_1\sqcup\cdots\sqcup M_q \to M |
103 | 2003 |
\] |
2004 |
of a marked $k$-ball $M$ |
|
2005 |
into small (marked and plain) $k$-balls $M_i$ and $X_j$, there is a |
|
2006 |
map from an appropriate subset (like a fibered product) |
|
2007 |
of |
|
2008 |
\[ |
|
2009 |
\cC(X_1)\times\cdots\times\cC(X_p) \times \cM(M_1)\times\cdots\times\cM(M_q) |
|
2010 |
\] |
|
2011 |
to $\cM(M)$, |
|
2012 |
and these various multifold composition maps satisfy an |
|
2013 |
operad-type strict associativity condition.} |
|
2014 |
||
423 | 2015 |
The above operad-like structure is analogous to the swiss cheese operad |
2016 |
\cite{MR1718089}. |
|
2017 |
||
2018 |
\medskip |
|
2019 |
||
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|
2020 |
We can define marked pinched products $\pi:E\to M$ of marked balls similarly to the |
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|
2021 |
plain ball case. A marked pinched product $\pi: E \to M$ is a pinched product (that is, locally modeled on degeneracy maps) which restricts to a map between the markings which is also a pinched product, and in a neighborhood of the markings is the product of the map between the markings with an interval. |
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|
2022 |
\nn{figure, 2 examples} |
423 | 2023 |
Note that a marked pinched product can be decomposed into either |
2024 |
two marked pinched products or a plain pinched product and a marked pinched product. |
|
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|
2025 |
\nn{should give figure} |
103 | 2026 |
|
423 | 2027 |
\begin{module-axiom}[Product (identity) morphisms] |
2028 |
For each pinched product $\pi:E\to M$, with $M$ a marked $k$-ball and $E$ a marked |
|
2029 |
$k{+}m$-ball ($m\ge 1$), |
|
424
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|
2030 |
there is a map $\pi^*:\cM(M)\to \cM(E)$. |
423 | 2031 |
These maps must satisfy the following conditions. |
2032 |
\begin{enumerate} |
|
2033 |
\item |
|
2034 |
If $\pi:E\to M$ and $\pi':E'\to M'$ are marked pinched products, and |
|
2035 |
if $f:M\to M'$ and $\tilde{f}:E \to E'$ are maps such that the diagram |
|
103 | 2036 |
\[ \xymatrix{ |
423 | 2037 |
E \ar[r]^{\tilde{f}} \ar[d]_{\pi} & E' \ar[d]^{\pi'} \\ |
103 | 2038 |
M \ar[r]^{f} & M' |
2039 |
} \] |
|
423 | 2040 |
commutes, then we have |
2041 |
\[ |
|
2042 |
\pi'^*\circ f = \tilde{f}\circ \pi^*. |
|
2043 |
\] |
|
2044 |
\item |
|
2045 |
Product morphisms are compatible with module composition and module action. |
|
2046 |
Let $\pi:E\to M$, $\pi_1:E_1\to M_1$, and $\pi_2:E_2\to M_2$ |
|
2047 |
be pinched products with $E = E_1\cup E_2$. |
|
2048 |
Let $a\in \cM(M)$, and let $a_i$ denote the restriction of $a$ to $M_i\sub M$. |
|
2049 |
Then |
|
2050 |
\[ |
|
2051 |
\pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) . |
|
2052 |
\] |
|
2053 |
Similarly, if $\rho:D\to X$ is a pinched product of plain balls and |
|
2054 |
$E = D\cup E_1$, then |
|
2055 |
\[ |
|
2056 |
\pi^*(a) = \rho^*(a')\bullet \pi_1^*(a_1), |
|
2057 |
\] |
|
2058 |
where $a'$ is the restriction of $a$ to $D$. |
|
2059 |
\item |
|
2060 |
Product morphisms are associative. |
|
2061 |
If $\pi:E\to M$ and $\rho:D\to E$ are marked pinched products then |
|
2062 |
\[ |
|
2063 |
\rho^*\circ\pi^* = (\pi\circ\rho)^* . |
|
2064 |
\] |
|
2065 |
\item |
|
2066 |
Product morphisms are compatible with restriction. |
|
2067 |
If we have a commutative diagram |
|
2068 |
\[ \xymatrix{ |
|
2069 |
D \ar@{^(->}[r] \ar[d]_{\rho} & E \ar[d]^{\pi} \\ |
|
2070 |
Y \ar@{^(->}[r] & M |
|
2071 |
} \] |
|
2072 |
such that $\rho$ and $\pi$ are pinched products, then |
|
2073 |
\[ |
|
2074 |
\res_D\circ\pi^* = \rho^*\circ\res_Y . |
|
2075 |
\] |
|
2076 |
($Y$ could be either a marked or plain ball.) |
|
2077 |
\end{enumerate} |
|
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|
2078 |
\end{module-axiom} |
103 | 2079 |
|
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|
2080 |
As in the $n$-category definition, once we have product morphisms we can define |
424
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|
2081 |
collar maps $\cM(M)\to \cM(M)$. |
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|
2082 |
Note that there are two cases: |
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|
2083 |
the collar could intersect the marking of the marked ball $M$, in which case |
6ebf92d2ccef
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|
2084 |
we use a product on a morphism of $\cM$; or the collar could be disjoint from the marking, |
6ebf92d2ccef
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|
2085 |
in which case we use a product on a morphism of $\cC$. |
6ebf92d2ccef
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|
2086 |
|
6ebf92d2ccef
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|
2087 |
In our example, elements $a$ of $\cM(M)$ maps to $T$, and $\pi^*(a)$ is the pullback of |
6ebf92d2ccef
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|
2088 |
$a$ along a map associated to $\pi$. |
6ebf92d2ccef
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|
2089 |
|
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|
2090 |
\medskip |
110 | 2091 |
|
917 | 2092 |
%There are two alternatives for the next axiom, according to whether we are defining |
2093 |
%modules for ordinary $n$-categories or $A_\infty$ $n$-categories. |
|
2094 |
%In the ordinary case we require |
|
2095 |
||
2096 |
The remaining module axioms are very similar to their counterparts in \S\ref{ss:n-cat-def}. |
|
103 | 2097 |
|
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|
2098 |
\begin{module-axiom}[Extended isotopy invariance in dimension $n$] |
918
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|
2099 |
\label{ei-module-axiom} |
916
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|
2100 |
Let $M$ be a marked $n$-ball, $b \in \cM(M)$, and $f: M\to M$ be a homeomorphism which |
7d398420577d
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diff
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|
2101 |
acts trivially on the restriction $\bd b$ of $b$ to $\bd M$. |
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|
2102 |
Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which |
7d398420577d
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diff
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|
2103 |
act trivially on $\bd b$. |
7d398420577d
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|
2104 |
Then $f(b) = b$. |
424
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|
2105 |
In addition, collar maps act trivially on $\cM(M)$. |
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|
2106 |
\end{module-axiom} |
103 | 2107 |
|
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|
2108 |
We emphasize that the $\bd M$ above (and below) means boundary in the marked $k$-ball sense. |
103 | 2109 |
In other words, if $M = (B, N)$ then we require only that isotopies are fixed |
2110 |
on $\bd B \setmin N$. |
|
2111 |
||
917 | 2112 |
\begin{module-axiom}[Splittings] |
2113 |
Let $c\in \cM_k(M)$, with $1\le k < n$. |
|
2114 |
Let $s = \{X_i\}$ be a splitting of M (so $M = \cup_i X_i$, and each $X_i$ is either a marked ball or a plain ball). |
|
2115 |
Let $\Homeo_\bd(M)$ denote homeomorphisms of $M$ which restrict to the identity on $\bd M$. |
|
2116 |
\begin{itemize} |
|
2117 |
\item (Alternative 1) Consider the set of homeomorphisms $g:M\to M$ such that $c$ splits along $g(s)$. |
|
2118 |
Then this subset of $\Homeo(M)$ is open and dense. |
|
2119 |
Furthermore, if $s$ restricts to a splitting $\bd s$ of $\bd M$, and if $\bd c$ splits along $\bd s$, then the |
|
2120 |
intersection of the set of such homeomorphisms $g$ with $\Homeo_\bd(M)$ is open and dense in $\Homeo_\bd(M)$. |
|
2121 |
\item (Alternative 2) Then there exists an embedded cell complex $S_c \sub M$, called the string locus of $c$, |
|
2122 |
such that if the splitting $s$ is transverse to $S_c$ then $c$ splits along $s$. |
|
2123 |
\end{itemize} |
|
2124 |
\end{module-axiom} |
|
2125 |
||
2126 |
We define the |
|
2127 |
category $\mbc$ of {\it marked $n$-balls with boundary conditions} as follows. |
|
2128 |
Its objects are pairs $(M, c)$, where $M$ is a marked $n$-ball and $c \in \cl\cM(\bd M)$ is the ``boundary condition". |
|
2129 |
The morphisms from $(M, c)$ to $(M', c')$, denoted $\Homeo(M; c \to M'; c')$, are |
|
2130 |
homeomorphisms $f:M\to M'$ such that $f|_{\bd M}(c) = c'$. |
|
2131 |
||
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|
2132 |
Let $\cS$ be a distributive symmetric monoidal category, and assume that $\cC$ is enriched in $\cS$. |
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|
2133 |
A $\cC$-module enriched in $\cS$ is defined analogously to \ref{axiom:enriched}. |
80fe92f8f81f
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|
2134 |
The top-dimensional part of the module $\cM_n$ is required to be a functor from $\mbc$ to $\cS$. |
80fe92f8f81f
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917
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|
2135 |
The top-dimensional gluing maps (module composition and $n$-category action) are $\cS$-maps whose |
80fe92f8f81f
finished updating module axioms (but have not done a proof-read)
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|
2136 |
domain is a direct sub of tensor products, as in \ref{axiom:enriched}. |
80fe92f8f81f
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|
2137 |
|
80fe92f8f81f
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|
2138 |
If $\cC$ is an $A_\infty$ $n$-category (see \ref{axiom:families}), we replace module axiom \ref{ei-module-axiom} |
80fe92f8f81f
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|
2139 |
with the following axiom. |
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917
diff
changeset
|
2140 |
Retain notation from \ref{axiom:families}. |
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|
2141 |
|
80fe92f8f81f
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917
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changeset
|
2142 |
\begin{module-axiom}[Families of homeomorphisms act in dimension $n$.] |
80fe92f8f81f
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Kevin Walker <kevin@canyon23.net>
parents:
917
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changeset
|
2143 |
For each pair of marked $n$-balls $M$ and $M'$ and each pair $c\in \cl{\cM}(\bd M)$ and $c'\in \cl{\cM}(\bd M')$ |
80fe92f8f81f
finished updating module axioms (but have not done a proof-read)
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917
diff
changeset
|
2144 |
we have an $\cS$-morphism |
103 | 2145 |
\[ |
918
80fe92f8f81f
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917
diff
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|
2146 |
\cJ(\Homeo(M;c \to M'; c')) \ot \cM(M; c) \to \cM(M'; c') . |
103 | 2147 |
\] |
918
80fe92f8f81f
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changeset
|
2148 |
Similarly, we have an $\cS$-morphism |
80fe92f8f81f
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parents:
917
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changeset
|
2149 |
\[ |
80fe92f8f81f
finished updating module axioms (but have not done a proof-read)
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parents:
917
diff
changeset
|
2150 |
\cJ(\Coll(M,c)) \ot \cM(M; c) \to \cM(M; c), |
80fe92f8f81f
finished updating module axioms (but have not done a proof-read)
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changeset
|
2151 |
\] |
80fe92f8f81f
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parents:
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changeset
|
2152 |
where $\Coll(M,c)$ denotes the space of collar maps. |
80fe92f8f81f
finished updating module axioms (but have not done a proof-read)
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parents:
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|
2153 |
These action maps are required to be associative up to coherent homotopy, |
424
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ncat.tex mostly module stuff
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423
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|
2154 |
and also compatible with composition (gluing) in the sense that |
437 | 2155 |
a diagram like the one in Theorem \ref{thm:CH} commutes. |
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|
2156 |
\end{module-axiom} |
103 | 2157 |
|
424
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|
2158 |
|
103 | 2159 |
\medskip |
102 | 2160 |
|
888
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|
2161 |
Note that the above axioms imply that an $n$-category module has the structure |
a0fd6e620926
Backed out changeset 7abe7642265e
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|
2162 |
of an $n{-}1$-category. |
104 | 2163 |
More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$, |
346
90e0c5e7ae07
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|
2164 |
where $X$ is a $k$-ball and in the product $X\times J$ we pinch |
104 | 2165 |
above the non-marked boundary component of $J$. |
200 | 2166 |
(More specifically, we collapse $X\times P$ to a single point, where |
2167 |
$P$ is the non-marked boundary component of $J$.) |
|
888
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|
2168 |
Then $\cE$ has the structure of an $n{-}1$-category. |
102 | 2169 |
|
105 | 2170 |
All marked $k$-balls are homeomorphic, unless $k = 1$ and our manifolds |
2171 |
are oriented or Spin (but not unoriented or $\text{Pin}_\pm$). |
|
2172 |
In this case ($k=1$ and oriented or Spin), there are two types |
|
2173 |
of marked 1-balls, call them left-marked and right-marked, |
|
2174 |
and hence there are two types of modules, call them right modules and left modules. |
|
2175 |
In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$), |
|
2176 |
there is no left/right module distinction. |
|
2177 |
||
130 | 2178 |
\medskip |
2179 |
||
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|
2180 |
We now give some examples of modules over ordinary and $A_\infty$ $n$-categories. |
224
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|
2181 |
|
225
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|
2182 |
\begin{example}[Examples from TQFTs] |
425 | 2183 |
\rm |
2184 |
Continuing Example \ref{ex:ncats-from-tqfts}, with $\cF$ a TQFT, $W$ an $n{-}j$-manifold, |
|
888
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|
2185 |
and $\cF(W)$ the $j$-category associated to $W$. |
425 | 2186 |
Let $Y$ be an $(n{-}j{+}1)$-manifold with $\bd Y = W$. |
2187 |
Define a $\cF(W)$ module $\cF(Y)$ as follows. |
|
2188 |
If $M = (B, N)$ is a marked $k$-ball with $k<j$ let |
|
2189 |
$\cF(Y)(M)\deq \cF((B\times W) \cup (N\times Y))$. |
|
2190 |
If $M = (B, N)$ is a marked $j$-ball and $c\in \cl{\cF(Y)}(\bd M)$ let |
|
2191 |
$\cF(Y)(M)\deq A_\cF((B\times W) \cup (N\times Y); c)$. |
|
225
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|
2192 |
\end{example} |
108 | 2193 |
|
448
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|
2194 |
\begin{example}[Examples from the blob complex] \label{bc-module-example} |
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|
2195 |
\rm |
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changeset
|
2196 |
In the previous example, we can instead define |
494
cb76847c439e
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changeset
|
2197 |
$\cF(Y)(M)\deq \bc_*((B\times W) \cup (N\times Y), c; \cF)$ (when $\dim(M) = n$) |
888
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|
2198 |
and get a module for the $A_\infty$ $n$-category associated to $\cF$ as in |
448
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|
2199 |
Example \ref{ex:blob-complexes-of-balls}. |
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2200 |
\end{example} |
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|
2201 |
|
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2202 |
|
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2203 |
\begin{example} |
425 | 2204 |
\rm |
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2205 |
Suppose $S$ is a topological space, with a subspace $T$. |
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2206 |
We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$ |
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2207 |
for $k<n$ the set $\pi_{\leq n}(S,T)(B,N)$ consists of all continuous maps of pairs |
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2208 |
$(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all |
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2209 |
such maps modulo homotopies fixed on $\bdy B \setminus N$. |
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|
2210 |
This is a module over the fundamental $n$-category $\pi_{\leq n}(S)$ of $S$, from Example \ref{ex:maps-to-a-space}. |
420 | 2211 |
\end{example} |
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2212 |
Modifications corresponding to Examples \ref{ex:maps-to-a-space-with-a-fiber} and |
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2213 |
\ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to |
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2214 |
Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains. |
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2215 |
|
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2216 |
|
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|
2217 |
|
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|
2218 |
|
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|
2219 |
|
324
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|
2220 |
\subsection{Modules as boundary labels (colimits for decorated manifolds)} |
112 | 2221 |
\label{moddecss} |
108 | 2222 |
|
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|
2223 |
Fix an ordinary $n$-category or $A_\infty$ $n$-category $\cC$. |
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2224 |
Let $W$ be a $k$-manifold ($k\le n$), |
143 | 2225 |
let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$, |
2226 |
and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$. |
|
2227 |
||
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|
2228 |
We will define a set $\cC(W, \cN)$ using a colimit construction very similar to |
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2229 |
the one appearing in \S \ref{ss:ncat_fields} above. |
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|
2230 |
(If $k = n$ and our $n$-categories are enriched, then |
108 | 2231 |
$\cC(W, \cN)$ will have additional structure; see below.) |
2232 |
||
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|
2233 |
Define a permissible decomposition of $W$ to be a map |
108 | 2234 |
\[ |
494
cb76847c439e
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|
2235 |
\left(\bigsqcup_a X_a\right) \sqcup \left(\bigsqcup_{i,b} M_{ib}\right) \to W, |
108 | 2236 |
\] |
494
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|
2237 |
where each $X_a$ is a plain $k$-ball disjoint, in $W$, from $\cup Y_i$, and |
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|
2238 |
each $M_{ib}$ is a marked $k$-ball intersecting $Y_i$ (once mapped into $W$), |
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|
2239 |
with $M_{ib}\cap Y_i$ being the marking, which extends to a ball decomposition in the sense of Definition \ref{defn:gluing-decomposition}. |
143 | 2240 |
(See Figure \ref{mblabel}.) |
435 | 2241 |
\begin{figure}[t] |
2242 |
\begin{equation*} |
|
286
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|
2243 |
\mathfig{.4}{ncat/mblabel} |
435 | 2244 |
\end{equation*} |
2245 |
\caption{A permissible decomposition of a manifold |
|
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|
2246 |
whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$. |
435 | 2247 |
Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel} |
2248 |
\end{figure} |
|
108 | 2249 |
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
2250 |
of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
|
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|
2251 |
This defines a partial ordering $\cell(W)$, which we will think of as a category. |
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|
2252 |
(The objects of $\cell(D)$ are permissible decompositions of $W$, and there is a unique |
108 | 2253 |
morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
2254 |
||
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|
2255 |
The collection of modules $\cN$ determines |
329
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|
2256 |
a functor $\psi_\cN$ from $\cell(W)$ to the category of sets |
108 | 2257 |
(possibly with additional structure if $k=n$). |
329
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|
2258 |
For a decomposition $x = (X_a, M_{ib})$ in $\cell(W)$, define $\psi_\cN(x)$ to be the subset |
108 | 2259 |
\[ |
191
8c2c330e87f2
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|
2260 |
\psi_\cN(x) \sub \left(\prod_a \cC(X_a)\right) \times \left(\prod_{ib} \cN_i(M_{ib})\right) |
108 | 2261 |
\] |
2262 |
such that the restrictions to the various pieces of shared boundaries amongst the |
|
2263 |
$X_a$ and $M_{ib}$ all agree. |
|
435 | 2264 |
(That is, the fibered product over the boundary restriction maps.) |
108 | 2265 |
If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$ |
2266 |
via the gluing (composition or action) maps from $\cC$ and the $\cN_i$. |
|
2267 |
||
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|
2268 |
We now define the set $\cC(W, \cN)$ to be the colimit of the functor $\psi_\cN$. |
435 | 2269 |
(As in \S\ref{ss:ncat-coend}, if $k=n$ we take a colimit in whatever |
2270 |
category we are enriching over, and if additionally we are in the $A_\infty$ case, |
|
2271 |
then we use a homotopy colimit.) |
|
2272 |
||
2273 |
\medskip |
|
108 | 2274 |
|
143 | 2275 |
If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define |
2276 |
$\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold |
|
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|
2277 |
$D\times Y_i \sub \bd(D\times W)$. |
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|
2278 |
It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$ |
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|
2279 |
has the structure of an $n{-}k$-category. |
144 | 2280 |
|
2281 |
\medskip |
|
2282 |
||
2283 |
We will use a simple special case of the above |
|
2284 |
construction to define tensor products |
|
2285 |
of modules. |
|
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a0fd6e620926
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|
2286 |
Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$. |
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|
2287 |
(If $k=1$ and our manifolds are oriented, then one should be |
144 | 2288 |
a left module and the other a right module.) |
2289 |
Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$. |
|
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|
2290 |
Define the tensor product $\cM_1 \tensor \cM_2$ to be the |
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a0fd6e620926
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|
2291 |
$n{-}1$-category associated as above to $J$ with its boundary labeled by $\cM_1$ and $\cM_2$. |
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|
2292 |
This of course depends (functorially) |
144 | 2293 |
on the choice of 1-ball $J$. |
105 | 2294 |
|
144 | 2295 |
We will define a more general self tensor product (categorified coend) below. |
2296 |
||
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2297 |
|
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|
2298 |
|
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|
2299 |
|
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|
2300 |
\subsection{Morphisms of modules} |
288 | 2301 |
\label{ss:module-morphisms} |
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2302 |
|
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|
2303 |
Modules are collections of functors together with some additional data, so we define morphisms |
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|
2304 |
of modules to be collections of natural transformations which are compatible with this |
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|
2305 |
additional data. |
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2306 |
|
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|
2307 |
More specifically, let $\cX$ and $\cY$ be $\cC$ modules, i.e.\ collections of functors |
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|
2308 |
$\{\cX_k\}$ and $\{\cY_k\}$, for $0\le k\le n$, from marked $k$-balls to sets |
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|
2309 |
as in Module Axiom \ref{module-axiom-funct}. |
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|
2310 |
A morphism $g:\cX\to\cY$ is a collection of natural transformations $g_k:\cX_k\to\cY_k$ |
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|
2311 |
satisfying: |
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|
2312 |
\begin{itemize} |
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|
2313 |
\item Each $g_k$ commutes with $\bd$. |
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|
2314 |
\item Each $g_k$ commutes with gluing (module composition and $\cC$ action). |
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|
2315 |
\item Each $g_k$ commutes with taking products. |
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|
2316 |
\item In the top dimension $k=n$, $g_n$ preserves whatever additional structure we are enriching over (e.g.\ vector |
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|
2317 |
spaces). |
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|
2318 |
In the $A_\infty$ case (e.g.\ enriching over chain complexes) $g_n$ should live in |
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|
2319 |
an appropriate derived hom space, as described below. |
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|
2320 |
\end{itemize} |
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|
2321 |
|
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|
2322 |
We will be mainly interested in the case $n=1$ and enriched over chain complexes, |
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|
2323 |
since this is the case that's relevant to the generalized Deligne conjecture of \S\ref{sec:deligne}. |
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|
2324 |
So we treat this case in more detail. |
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|
2325 |
|
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|
2326 |
First we explain the remark about derived hom above. |
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|
2327 |
Let $L$ be a marked 1-ball and let $\cl{\cX}(L)$ denote the local homotopy colimit construction |
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|
2328 |
associated to $L$ by $\cX$ and $\cC$. |
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|
2329 |
(See \S \ref{ss:ncat_fields} and \S \ref{moddecss}.) |
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|
2330 |
Define $\cl{\cY}(L)$ similarly. |
879 | 2331 |
For $K$ an unmarked 1-ball let $\cl{\cC}(K)$ denote the local homotopy colimit |
546
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|
2332 |
construction associated to $K$ by $\cC$. |
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|
2333 |
Then we have an injective gluing map |
261
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|
2334 |
\[ |
546
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|
2335 |
\gl: \cl{\cX}(L) \ot \cl{\cC}(K) \to \cl{\cX}(L\cup K) |
261
1c408505c9f5
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|
2336 |
\] |
546
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|
2337 |
which is also a chain map. |
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|
2338 |
(For simplicity we are suppressing mention of boundary conditions on the unmarked |
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|
2339 |
boundary components of the 1-balls.) |
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|
2340 |
We define $\hom_\cC(\cX \to \cY)$ to be a collection of (graded linear) natural transformations |
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|
2341 |
$g: \cl{\cX}(L)\to \cl{\cY}(L)$ such that the following diagram commutes for all $L$ and $K$: |
262
3278eafef668
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|
2342 |
\[ \xymatrix{ |
546
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2343 |
\cl{\cX}(L) \ot \cl{\cC}(K) \ar[r]^{\gl} \ar[d]_{g\ot \id} & \cl{\cX}(L\cup K) \ar[d]^{g}\\ |
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\cl{\cY}(L) \ot \cl{\cC}(K) \ar[r]^{\gl} & \cl{\cY}(L\cup K) |
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2345 |
} \] |
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2346 |
|
546
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2347 |
The usual differential on graded linear maps between chain complexes induces a differential |
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2348 |
on $\hom_\cC(\cX \to \cY)$, giving it the structure of a chain complex. |
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2349 |
|
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2350 |
Let $\cZ$ be another $\cC$ module. |
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2351 |
We define a chain map |
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2352 |
\[ |
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2353 |
a: \hom_\cC(\cX \to \cY) \ot (\cX \ot_\cC \cZ) \to \cY \ot_\cC \cZ |
386 | 2354 |
\] |
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2355 |
as follows. |
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|
2356 |
Recall that the tensor product $\cX \ot_\cC \cZ$ depends on a choice of interval $J$, labeled |
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2357 |
by $\cX$ on one boundary component and $\cZ$ on the other. |
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|
2358 |
Because we are using the {\it local} homotopy colimit, any generator |
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|
2359 |
$D\ot x\ot \bar{c}\ot z$ of $\cX \ot_\cC \cZ$ can be written (perhaps non-uniquely) as a gluing |
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|
2360 |
$(D'\ot x \ot \bar{c}') \bullet (D''\ot \bar{c}''\ot z)$, for some decomposition $J = L'\cup L''$ |
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2361 |
and with $D'\ot x \ot \bar{c}'$ a generator of $\cl{\cX}(L')$ and |
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|
2362 |
$D''\ot \bar{c}''\ot z$ a generator of $\cl{\cZ}(L'')$. |
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|
2363 |
(Such a splitting exists because the blob diagram $D$ can be split into left and right halves, |
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2364 |
since no blob can include both the leftmost and rightmost intervals in the underlying decomposition. |
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2365 |
This step would fail if we were using the usual hocolimit instead of the local hocolimit.) |
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2366 |
We now define |
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|
2367 |
\[ |
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2368 |
a: g\ot (D\ot x\ot \bar{c}\ot z) \mapsto g(D'\ot x \ot \bar{c}')\bullet (D''\ot \bar{c}''\ot z) . |
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|
2369 |
\] |
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2370 |
This does not depend on the choice of splitting $D = D'\bullet D''$ because $g$ commutes with gluing. |
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2371 |
|
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2372 |
|
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2373 |
|
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2374 |
|
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2375 |
\subsection{The \texorpdfstring{$n{+}1$}{n+1}-category of sphere modules} |
218 | 2376 |
\label{ssec:spherecat} |
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2377 |
|
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|
2378 |
In this subsection we define $n{+}1$-categories $\cS$ of ``sphere modules". |
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|
2379 |
The objects are $n$-categories, the $k$-morphisms are $k{-}1$-sphere modules for $1\le k \le n$, |
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858b80dfa05c
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2380 |
and the $n{+}1$-morphisms are intertwiners. |
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2381 |
With future applications in mind, we treat simultaneously the big category |
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|
2382 |
of all $n$-categories and all sphere modules and also subcategories thereof. |
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2383 |
When $n=1$ this is closely related to familiar $2$-categories consisting of |
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2384 |
algebras, bimodules and intertwiners (or a subcategory of that). |
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|
2385 |
The sphere module $n{+}1$-category is a natural generalization of the |
811
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|
2386 |
algebra-bimodule-intertwiner 2-category to higher dimensions. |
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|
2387 |
|
866 | 2388 |
Another possible name for this $n{+}1$-category is the $n{+}1$-category of defects. |
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|
2389 |
The $n$-categories are thought of as representing field theories, and the |
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|
2390 |
$0$-sphere modules are codimension 1 defects between adjacent theories. |
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|
2391 |
In general, $m$-sphere modules are codimension $m{+}1$ defects; |
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|
2392 |
the link of such a defect is an $m$-sphere decorated with defects of smaller codimension. |
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|
2393 |
|
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|
2394 |
\medskip |
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|
2395 |
|
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|
2396 |
While it is appropriate to call an $S^0$ module a bimodule, |
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|
2397 |
this is much less true for higher dimensional spheres, |
327 | 2398 |
so we prefer the term ``sphere module" for the general case. |
144 | 2399 |
|
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|
2400 |
For simplicity, we will assume that $n$-categories are enriched over $\c$-vector spaces. |
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|
2401 |
|
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|
2402 |
The $0$- through $n$-dimensional parts of $\cS$ are various sorts of modules, and we describe |
205 | 2403 |
these first. |
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|
2404 |
The $n{+}1$-dimensional part of $\cS$ consists of intertwiners |
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|
2405 |
of $1$-category modules associated to decorated $n$-balls. |
205 | 2406 |
We will see below that in order for these $n{+}1$-morphisms to satisfy all of |
398
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|
2407 |
the axioms of an $n{+}1$-category (in particular, duality requirements), we will have to assume |
205 | 2408 |
that our $n$-categories and modules have non-degenerate inner products. |
2409 |
(In other words, we need to assume some extra duality on the $n$-categories and modules.) |
|
2410 |
||
2411 |
\medskip |
|
2412 |
||
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|
2413 |
Our first task is to define an $n$-category $m$-sphere module, for $0\le m \le n-1$. |
205 | 2414 |
These will be defined in terms of certain classes of marked balls, very similarly |
2415 |
to the definition of $n$-category modules above. |
|
2416 |
(This, in turn, is very similar to our definition of $n$-category.) |
|
2417 |
Because of this similarity, we only sketch the definitions below. |
|
2418 |
||
327 | 2419 |
We start with $0$-sphere modules, which also could reasonably be called (categorified) bimodules. |
205 | 2420 |
(For $n=1$ they are precisely bimodules in the usual, uncategorified sense.) |
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|
2421 |
We prefer the more awkward term ``0-sphere module" to emphasize the analogy |
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|
2422 |
with the higher sphere modules defined below. |
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|
2423 |
|
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|
2424 |
Define a $0$-marked $k$-ball, $1\le k \le n$, to be a pair $(X, M)$ homeomorphic to the standard |
327 | 2425 |
$(B^k, B^{k-1})$. |
209 | 2426 |
See Figure \ref{feb21a}. |
205 | 2427 |
Another way to say this is that $(X, M)$ is homeomorphic to $B^{k-1}\times([-1,1], \{0\})$. |
2428 |
||
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|
2429 |
\begin{figure}[t] |
387
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|
2430 |
$$\tikz[baseline,line width=2pt]{\draw[blue] (-2,0)--(2,0); \fill[red] (0,0) circle (0.1);} \qquad \qquad \tikz[baseline,line width=2pt]{\draw[blue][fill=blue!30!white] (0,0) circle (2 and 1); \draw[red] (0,1)--(0,-1);}$$ |
209 | 2431 |
\caption{0-marked 1-ball and 0-marked 2-ball} |
2432 |
\label{feb21a} |
|
2433 |
\end{figure} |
|
2434 |
||
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|
2435 |
The $0$-marked balls can be cut into smaller balls in various ways. |
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|
2436 |
We only consider those decompositions in which the smaller balls are either |
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|
2437 |
$0$-marked (i.e. intersect the $0$-marking of the large ball in a disc) |
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|
2438 |
or plain (don't intersect the $0$-marking of the large ball). |
879 | 2439 |
We can also take the boundary of a $0$-marked ball, which is a $0$-marked sphere. |
205 | 2440 |
|
2441 |
Fix $n$-categories $\cA$ and $\cB$. |
|
327 | 2442 |
These will label the two halves of a $0$-marked $k$-ball. |
205 | 2443 |
|
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|
2444 |
An $n$-category $0$-sphere module $\cM$ over the $n$-categories $\cA$ and $\cB$ is |
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|
2445 |
a collection of functors $\cM_k$ from the category |
327 | 2446 |
of $0$-marked $k$-balls, $1\le k \le n$, |
205 | 2447 |
(with the two halves labeled by $\cA$ and $\cB$) to the category of sets. |
2448 |
If $k=n$ these sets should be enriched to the extent $\cA$ and $\cB$ are. |
|
327 | 2449 |
Given a decomposition of a $0$-marked $k$-ball $X$ into smaller balls $X_i$, we have |
205 | 2450 |
morphism sets $\cA_k(X_i)$ (if $X_i$ lies on the $\cA$-labeled side) |
2451 |
or $\cB_k(X_i)$ (if $X_i$ lies on the $\cB$-labeled side) |
|
2452 |
or $\cM_k(X_i)$ (if $X_i$ intersects the marking and is therefore a smaller 0-marked ball). |
|
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|
2453 |
Corresponding to this decomposition we have a composition (or ``gluing") map |
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|
2454 |
from the product (fibered over the boundary data) of these various sets into $\cM_k(X)$. |
205 | 2455 |
|
2456 |
\medskip |
|
107 | 2457 |
|
327 | 2458 |
Part of the structure of an $n$-category 0-sphere module $\cM$ is captured by saying it is |
206 | 2459 |
a collection $\cD^{ab}$ of $n{-}1$-categories, indexed by pairs $(a, b)$ of objects (0-morphisms) |
2460 |
of $\cA$ and $\cB$. |
|
2461 |
Let $J$ be some standard 0-marked 1-ball (i.e.\ an interval with a marked point in its interior). |
|
2462 |
Given a $j$-ball $X$, $0\le j\le n-1$, we define |
|
2463 |
\[ |
|
2464 |
\cD(X) \deq \cM(X\times J) . |
|
2465 |
\] |
|
2466 |
The product is pinched over the boundary of $J$. |
|
327 | 2467 |
The set $\cD$ breaks into ``blocks" according to the restrictions to the pinched points of $X\times J$ |
209 | 2468 |
(see Figure \ref{feb21b}). |
206 | 2469 |
These restrictions are 0-morphisms $(a, b)$ of $\cA$ and $\cB$. |
107 | 2470 |
|
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|
2471 |
\begin{figure}[t] \centering |
367 | 2472 |
\begin{tikzpicture}[blue,line width=2pt] |
2473 |
\draw (0,1) -- (0,-1) node[below] {$X$}; |
|
2474 |
||
2475 |
\draw (2,0) -- (4,0) node[below] {$J$}; |
|
2476 |
\fill[red] (3,0) circle (0.1); |
|
2477 |
||
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|
2478 |
\draw[fill=blue!30!white] (6,0) node(a) {} arc (135:90:4) node(top) {} arc (90:45:4) node(b) {} arc (-45:-90:4) node(bottom) {} arc(-90:-135:4); |
367 | 2479 |
\draw[red] (top.center) -- (bottom.center); |
2480 |
\fill (a) circle (0.1) node[left] {\color{green!50!brown} $a$}; |
|
2481 |
\fill (b) circle (0.1) node[right] {\color{green!50!brown} $b$}; |
|
2482 |
||
2483 |
\path (bottom) node[below]{$X \times J$}; |
|
2484 |
||
2485 |
\end{tikzpicture} |
|
209 | 2486 |
\caption{The pinched product $X\times J$} |
2487 |
\label{feb21b} |
|
2488 |
\end{figure} |
|
2489 |
||
206 | 2490 |
More generally, consider an interval with interior marked points, and with the complements |
2491 |
of these points labeled by $n$-categories $\cA_i$ ($0\le i\le l$) and the marked points labeled |
|
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|
2492 |
by $\cA_i$-$\cA_{i+1}$ 0-sphere modules $\cM_i$. |
209 | 2493 |
(See Figure \ref{feb21c}.) |
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|
2494 |
To this data we can apply the coend construction as in \S\ref{moddecss} above |
327 | 2495 |
to obtain an $\cA_0$-$\cA_l$ $0$-sphere module and, forgetfully, an $n{-}1$-category. |
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|
2496 |
This amounts to a definition of taking tensor products of $0$-sphere modules over $n$-categories. |
205 | 2497 |
|
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changeset
|
2498 |
\begin{figure}[t] \centering |
367 | 2499 |
\begin{tikzpicture}[baseline,line width = 2pt] |
2500 |
\draw[blue] (0,0) -- (6,0); |
|
2501 |
\foreach \x/\n in {0.5/0,1.5/1,3/2,4.5/3,5.5/4} { |
|
2502 |
\path (\x,0) node[below] {\color{green!50!brown}$\cA_{\n}$}; |
|
2503 |
} |
|
2504 |
\foreach \x/\n in {1/0,2/1,4/2,5/3} { |
|
2505 |
\fill[red] (\x,0) circle (0.1) node[above] {\color{green!50!brown}$\cM_{\n}$}; |
|
2506 |
} |
|
2507 |
\end{tikzpicture} |
|
2508 |
\qquad |
|
2509 |
\qquad |
|
2510 |
\begin{tikzpicture}[baseline,line width = 2pt] |
|
2511 |
\draw[blue] (0,0) circle (2); |
|
2512 |
\foreach \q/\n in {-45/0,90/1,180/2} { |
|
2513 |
\path (\q:2.4) node {\color{green!50!brown}$\cA_{\n}$}; |
|
2514 |
} |
|
2515 |
\foreach \q/\n in {60/0,120/1,-120/2} { |
|
2516 |
\fill[red] (\q:2) circle (0.1); |
|
2517 |
\path (\q:2.4) node {\color{green!50!brown}$\cM_{\n}$}; |
|
2518 |
} |
|
2519 |
\end{tikzpicture} |
|
209 | 2520 |
\caption{Marked and labeled 1-manifolds} |
2521 |
\label{feb21c} |
|
2522 |
\end{figure} |
|
2523 |
||
206 | 2524 |
We could also similarly mark and label a circle, obtaining an $n{-}1$-category |
2525 |
associated to the marked and labeled circle. |
|
209 | 2526 |
(See Figure \ref{feb21c}.) |
206 | 2527 |
If the circle is divided into two intervals, we can think of this $n{-}1$-category |
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changeset
|
2528 |
as the 2-sided tensor product of the two 0-sphere modules associated to the two intervals. |
206 | 2529 |
|
2530 |
\medskip |
|
2531 |
||
2532 |
Next we define $n$-category 1-sphere modules. |
|
2533 |
These are just representations of (modules for) $n{-}1$-categories associated to marked and labeled |
|
2534 |
circles (1-spheres) which we just introduced. |
|
2535 |
||
2536 |
Equivalently, we can define 1-sphere modules in terms of 1-marked $k$-balls, $2\le k\le n$. |
|
2537 |
Fix a marked (and labeled) circle $S$. |
|
209 | 2538 |
Let $C(S)$ denote the cone of $S$, a marked 2-ball (Figure \ref{feb21d}). |
439
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changeset
|
2539 |
%\nn{I need to make up my mind whether marked things are always labeled too. |
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|
2540 |
%For the time being, let's say they are.} |
207 | 2541 |
A 1-marked $k$-ball is anything homeomorphic to $B^j \times C(S)$, $0\le j\le n-2$, |
2542 |
where $B^j$ is the standard $j$-ball. |
|
399 | 2543 |
A 1-marked $k$-ball can be decomposed in various ways into smaller balls, which are either |
2544 |
(a) smaller 1-marked $k$-balls, (b) 0-marked $k$-balls, or (c) plain $k$-balls. |
|
560 | 2545 |
(See Figure \ref{subdividing1marked}.) |
207 | 2546 |
We now proceed as in the above module definitions. |
2547 |
||
530
b236746e8e4d
futzing with figures (\begin{center|equation} to \centering)
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diff
changeset
|
2548 |
\begin{figure}[t] \centering |
367 | 2549 |
\begin{tikzpicture}[baseline,line width = 2pt] |
387
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|
2550 |
\draw[blue][fill=blue!15!white] (0,0) circle (2); |
367 | 2551 |
\fill[red] (0,0) circle (0.1); |
2552 |
\foreach \qm/\qa/\n in {70/-30/0, 120/95/1, -120/180/2} { |
|
2553 |
\draw[red] (0,0) -- (\qm:2); |
|
2554 |
\path (\qa:1) node {\color{green!50!brown} $\cA_\n$}; |
|
2555 |
\path (\qm+20:2.5) node(M\n) {\color{green!50!brown} $\cM_\n$}; |
|
2556 |
\draw[line width=1pt, green!50!brown, ->] (M\n.\qm+135) to[out=\qm+135,in=\qm+90] (\qm+5:1.3); |
|
2557 |
} |
|
2558 |
\end{tikzpicture} |
|
557 | 2559 |
\caption{Cone on a marked circle, the prototypical 1-marked ball} |
209 | 2560 |
\label{feb21d} |
2561 |
\end{figure} |
|
2562 |
||
560 | 2563 |
\begin{figure}[t] \centering |
2564 |
\begin{tikzpicture}[baseline,line width = 2pt] |
|
2565 |
\draw[blue][fill=blue!15!white] (0,0) circle (2); |
|
2566 |
\fill[red] (0,0) circle (0.1); |
|
2567 |
\foreach \qm/\qa/\n in {70/-30/0, 120/95/1, -120/180/2} { |
|
2568 |
\draw[red] (0,0) -- (\qm:2); |
|
2569 |
% \path (\qa:1) node {\color{green!50!brown} $\cA_\n$}; |
|
2570 |
% \path (\qm+20:2.5) node(M\n) {\color{green!50!brown} $\cM_\n$}; |
|
2571 |
% \draw[line width=1pt, green!50!brown, ->] (M\n.\qm+135) to[out=\qm+135,in=\qm+90] (\qm+5:1.3); |
|
2572 |
} |
|
2573 |
||
2574 |
||
2575 |
\begin{scope}[black, thin] |
|
2576 |
\clip (0,0) circle (2); |
|
2577 |
\draw (0:1) -- (90:1) -- (180:1) -- (270:1) -- cycle; |
|
2578 |
\draw (90:1) -- (90:2.1); |
|
2579 |
\draw (180:1) -- (180:2.1); |
|
2580 |
\draw (270:1) -- (270:2.1); |
|
2581 |
\draw (0:1) -- (15:2.1); |
|
2582 |
\draw (0:1) -- (315:1.5) -- (270:1); |
|
2583 |
\draw (315:1.5) -- (315:2.1); |
|
2584 |
\end{scope} |
|
2585 |
||
2586 |
\node(0marked) at (2.5,2.25) {$0$-marked ball}; |
|
2587 |
\node(1marked) at (3.5,1) {$1$-marked ball}; |
|
2588 |
\node(plain) at (3,-1) {plain ball}; |
|
2589 |
\draw[line width=1pt, green!50!brown, ->] (0marked.270) to[out=270,in=45] (50:1.1); |
|
2590 |
\draw[line width=1pt, green!50!brown, ->] (1marked.225) to[out=270,in=45] (0.4,0.1); |
|
2591 |
\draw[line width=1pt, green!50!brown, ->] (plain.90) to[out=135,in=45] (-45:1); |
|
2592 |
||
2593 |
\end{tikzpicture} |
|
2594 |
\caption{Subdividing a $1$-marked ball into plain, $0$-marked and $1$-marked balls.} |
|
2595 |
\label{subdividing1marked} |
|
2596 |
\end{figure} |
|
2597 |
||
207 | 2598 |
A $n$-category 1-sphere module is, among other things, an $n{-}2$-category $\cD$ with |
2599 |
\[ |
|
2600 |
\cD(X) \deq \cM(X\times C(S)) . |
|
2601 |
\] |
|
2602 |
The product is pinched over the boundary of $C(S)$. |
|
2603 |
$\cD$ breaks into ``blocks" according to the restriction to the |
|
2604 |
image of $\bd C(S) = S$ in $X\times C(S)$. |
|
2605 |
||
2606 |
More generally, consider a 2-manifold $Y$ |
|
2607 |
(e.g.\ 2-ball or 2-sphere) marked by an embedded 1-complex $K$. |
|
2608 |
The components of $Y\setminus K$ are labeled by $n$-categories, |
|
2609 |
the edges of $K$ are labeled by 0-sphere modules, |
|
2610 |
and the 0-cells of $K$ are labeled by 1-sphere modules. |
|
2611 |
We can now apply the coend construction and obtain an $n{-}2$-category. |
|
398
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|
2612 |
If $Y$ has boundary then this $n{-}2$-category is a module for the $n{-}1$-category |
207 | 2613 |
associated to the (marked, labeled) boundary of $Y$. |
2614 |
In particular, if $\bd Y$ is a 1-sphere then we get a 1-sphere module as defined above. |
|
2615 |
||
2616 |
\medskip |
|
2617 |
||
2618 |
It should now be clear how to define $n$-category $m$-sphere modules for $0\le m \le n-1$. |
|
2619 |
For example, there is an $n{-}2$-category associated to a marked, labeled 2-sphere, |
|
208 | 2620 |
and a 2-sphere module is a representation of such an $n{-}2$-category. |
207 | 2621 |
|
2622 |
\medskip |
|
2623 |
||
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|
2624 |
We can now define the $n$-or-less-dimensional part of our $n{+}1$-category $\cS$. |
439
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435
diff
changeset
|
2625 |
Choose some collection of $n$-categories, then choose some collections of 0-sphere modules between |
207 | 2626 |
these $n$-categories, then choose some collection of 1-sphere modules for the various |
439
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435
diff
changeset
|
2627 |
possible marked 1-spheres labeled by the $n$-categories and 0-sphere modules, and so on. |
207 | 2628 |
Let $L_i$ denote the collection of $i{-}1$-sphere modules we have chosen. |
2629 |
(For convenience, we declare a $(-1)$-sphere module to be an $n$-category.) |
|
2630 |
There is a wide range of possibilities. |
|
398
2a9c637182f0
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393
diff
changeset
|
2631 |
The set $L_0$ could contain infinitely many $n$-categories or just one. |
439
10f0f68cafb4
mostly (entirely?) ncat revisions
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parents:
435
diff
changeset
|
2632 |
For each pair of $n$-categories in $L_0$, $L_1$ could contain no 0-sphere modules at all or |
207 | 2633 |
it could contain several. |
208 | 2634 |
The only requirement is that each $k$-sphere module be a module for a $k$-sphere $n{-}k$-category |
2635 |
constructed out of labels taken from $L_j$ for $j<k$. |
|
2636 |
||
859 | 2637 |
We remind the reader again that $\cS = \cS_{\{L_i\}, \{z_Y\}}$ depends on |
2638 |
the choice of $L_i$ above as well as the choice of |
|
2639 |
families of inner products below. |
|
2640 |
||
398
2a9c637182f0
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changeset
|
2641 |
We now define $\cS(X)$, for $X$ a ball of dimension at most $n$, to be the set of all |
208 | 2642 |
cell-complexes $K$ embedded in $X$, with the codimension-$j$ parts of $(X, K)$ labeled |
2643 |
by elements of $L_j$. |
|
2644 |
As described above, we can think of each decorated $k$-ball as defining a $k{-}1$-sphere module |
|
2645 |
for the $n{-}k{+}1$-category associated to its decorated boundary. |
|
2646 |
Thus the $k$-morphisms of $\cS$ (for $k\le n$) can be thought |
|
2647 |
of as $n$-category $k{-}1$-sphere modules |
|
2648 |
(generalizations of bimodules). |
|
387
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diff
changeset
|
2649 |
On the other hand, we can equally well think of the $k$-morphisms as decorations on $k$-balls, |
398
2a9c637182f0
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diff
changeset
|
2650 |
and from this point of view it is clear that they satisfy all of the axioms of an |
208 | 2651 |
$n{+}1$-category. |
2652 |
(All of the axioms for the less-than-$n{+}1$-dimensional part of an $n{+}1$-category, that is.) |
|
2653 |
||
2654 |
\medskip |
|
2655 |
||
2656 |
Next we define the $n{+}1$-morphisms of $\cS$. |
|
387
f0518720227a
sphere modules (in progress)
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386
diff
changeset
|
2657 |
The construction of the 0- through $n$-morphisms was easy and tautological, but the |
398
2a9c637182f0
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393
diff
changeset
|
2658 |
$n{+}1$-morphisms will require some effort and combinatorial topology, as well as additional |
770
032d3c2b2a89
added remark about defect categories; tweaked sphere cat intro
Kevin Walker <kevin@canyon23.net>
parents:
766
diff
changeset
|
2659 |
duality assumptions on the lower morphisms. |
032d3c2b2a89
added remark about defect categories; tweaked sphere cat intro
Kevin Walker <kevin@canyon23.net>
parents:
766
diff
changeset
|
2660 |
These are required because we define the spaces of $n{+}1$-morphisms by |
032d3c2b2a89
added remark about defect categories; tweaked sphere cat intro
Kevin Walker <kevin@canyon23.net>
parents:
766
diff
changeset
|
2661 |
making arbitrary choices of incoming and outgoing boundaries for each $n$-ball. |
858
1fc5fff34251
typos, not from referee rpt
Kevin Walker <kevin@canyon23.net>
parents:
857
diff
changeset
|
2662 |
The additional duality assumptions are needed to prove independence of our definition from these choices. |
208 | 2663 |
|
387
f0518720227a
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386
diff
changeset
|
2664 |
Let $X$ be an $n{+}1$-ball, and let $c$ be a decoration of its boundary |
f0518720227a
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386
diff
changeset
|
2665 |
by a cell complex labeled by 0- through $n$-morphisms, as above. |
859 | 2666 |
Choose an $n{-}1$-sphere $E\sub \bd X$, transverse to $c$, which divides |
387
f0518720227a
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386
diff
changeset
|
2667 |
$\bd X$ into ``incoming" and ``outgoing" boundary $\bd_-X$ and $\bd_+X$. |
f0518720227a
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386
diff
changeset
|
2668 |
Let $E_c$ denote $E$ decorated by the restriction of $c$ to $E$. |
f0518720227a
sphere modules (in progress)
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diff
changeset
|
2669 |
Recall from above the associated 1-category $\cS(E_c)$. |
f0518720227a
sphere modules (in progress)
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386
diff
changeset
|
2670 |
We can also have $\cS(E_c)$ modules $\cS(\bd_-X_c)$ and $\cS(\bd_+X_c)$. |
f0518720227a
sphere modules (in progress)
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386
diff
changeset
|
2671 |
Define |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
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386
diff
changeset
|
2672 |
\[ |
f0518720227a
sphere modules (in progress)
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386
diff
changeset
|
2673 |
\cS(X; c; E) \deq \hom_{\cS(E_c)}(\cS(\bd_-X_c), \cS(\bd_+X_c)) . |
f0518720227a
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diff
changeset
|
2674 |
\] |
208 | 2675 |
|
439
10f0f68cafb4
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parents:
435
diff
changeset
|
2676 |
We will show that if the sphere modules are equipped with a ``compatible family of |
10f0f68cafb4
mostly (entirely?) ncat revisions
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parents:
435
diff
changeset
|
2677 |
non-degenerate inner products", then there is a coherent family of isomorphisms |
387
f0518720227a
sphere modules (in progress)
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386
diff
changeset
|
2678 |
$\cS(X; c; E) \cong \cS(X; c; E')$ for all pairs of choices $E$ and $E'$. |
439
10f0f68cafb4
mostly (entirely?) ncat revisions
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parents:
435
diff
changeset
|
2679 |
This will allow us to define $\cS(X; c)$ independently of the choice of $E$. |
208 | 2680 |
|
390
027bfdae3098
define compatible familty of non-degenerate IPs
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387
diff
changeset
|
2681 |
First we must define ``inner product", ``non-degenerate" and ``compatible". |
837 | 2682 |
Let $Y$ be a decorated $n$-ball, and $\ol{Y}$ its mirror image. |
387
f0518720227a
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386
diff
changeset
|
2683 |
(We assume we are working in the unoriented category.) |
f0518720227a
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diff
changeset
|
2684 |
Let $Y\cup\ol{Y}$ denote the decorated $n$-sphere obtained by gluing $Y$ and $\ol{Y}$ |
f0518720227a
sphere modules (in progress)
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parents:
386
diff
changeset
|
2685 |
along their common boundary. |
f0518720227a
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386
diff
changeset
|
2686 |
An {\it inner product} on $\cS(Y)$ is a dual vector |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2687 |
\[ |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2688 |
z_Y : \cS(Y\cup\ol{Y}) \to \c. |
f0518720227a
sphere modules (in progress)
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parents:
386
diff
changeset
|
2689 |
\] |
f0518720227a
sphere modules (in progress)
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parents:
386
diff
changeset
|
2690 |
We will also use the notation |
f0518720227a
sphere modules (in progress)
Kevin Walker <kevin@canyon23.net>
parents:
386
diff
changeset
|
2691 |
\[ |
857
4ad6325c7c7d
remove bar per referee (minor)
Kevin Walker <kevin@canyon23.net>
parents:
855
diff
changeset
|
2692 |
\langle a, b\rangle \deq z_Y(a\bullet b) \in \c . |
387
f0518720227a
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diff
changeset
|
2693 |
\] |
390
027bfdae3098
define compatible familty of non-degenerate IPs
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diff
changeset
|
2694 |
An inner product induces a linear map |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2695 |
\begin{eqnarray*} |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2696 |
\varphi: \cS(Y) &\to& \cS(Y)^* \\ |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2697 |
a &\mapsto& \langle a, \cdot \rangle |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2698 |
\end{eqnarray*} |
027bfdae3098
define compatible familty of non-degenerate IPs
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parents:
387
diff
changeset
|
2699 |
which satisfies, for all morphisms $e$ of $\cS(\bd Y)$, |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2700 |
\[ |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2701 |
\varphi(ae)(b) = \langle ae, b \rangle = z_Y(a\bullet e\bullet b) = |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2702 |
\langle a, eb \rangle = \varphi(a)(eb) . |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
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diff
changeset
|
2703 |
\] |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2704 |
In other words, $\varphi$ is a map of $\cS(\bd Y)$ modules. |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
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diff
changeset
|
2705 |
An inner product is {\it non-degenerate} if $\varphi$ is an isomorphism. |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2706 |
This implies that $\cS(Y; c)$ is finite dimensional for all boundary conditions $c$. |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2707 |
(One can think of these inner products as giving some duality in dimension $n{+}1$; |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2708 |
heretofore we have only assumed duality in dimensions 0 through $n$.) |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2709 |
|
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2710 |
Next we define compatibility. |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2711 |
Let $Y = Y_1\cup Y_2$ with $D = Y_1\cap Y_2$. |
398
2a9c637182f0
edits to sphere-modules stuff: some todos added
Scott Morrison <scott@tqft.net>
parents:
393
diff
changeset
|
2712 |
Let $X_1$ and $X_2$ be the two components of $Y\times I$ cut along |
2a9c637182f0
edits to sphere-modules stuff: some todos added
Scott Morrison <scott@tqft.net>
parents:
393
diff
changeset
|
2713 |
$D\times I$, in both cases using the pinched product. |
390
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2714 |
(Here we are overloading notation and letting $D$ denote both a decorated and an undecorated |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2715 |
manifold.) |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2716 |
We have $\bd X_i = Y_i \cup \ol{Y}_i \cup (D\times I)$ |
393 | 2717 |
(see Figure \ref{jun23a}). |
2718 |
\begin{figure}[t] |
|
2719 |
\begin{equation*} |
|
497 | 2720 |
\mathfig{.6}{ncat/YxI-sliced} |
393 | 2721 |
\end{equation*} |
2722 |
\caption{$Y\times I$ sliced open} |
|
2723 |
\label{jun23a} |
|
2724 |
\end{figure} |
|
390
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2725 |
Given $a_i\in \cS(Y_i)$, $b_i\in \cS(\ol{Y}_i)$ and $v\in\cS(D\times I)$ |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2726 |
which agree on their boundaries, we can evaluate |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2727 |
\[ |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2728 |
z_{Y_i}(a_i\bullet b_i\bullet v) \in \c . |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2729 |
\] |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2730 |
(This requires a choice of homeomorphism $Y_i \cup \ol{Y}_i \cup (D\times I) \cong |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2731 |
Y_i \cup \ol{Y}_i$, but the value of $z_{Y_i}$ is independent of this choice.) |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2732 |
We can think of $z_{Y_i}$ as giving a function |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2733 |
\[ |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2734 |
\psi_i : \cS(Y_i) \ot \cS(\ol{Y}_i) \to \cS(D\times I)^* |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2735 |
\stackrel{\varphi\inv}{\longrightarrow} \cS(D\times I) . |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2736 |
\] |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2737 |
We can now finally define a family of inner products to be {\it compatible} if |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2738 |
for all decompositions $Y = Y_1\cup Y_2$ as above and all $a_i\in \cS(Y_i)$, $b_i\in \cS(\ol{Y}_i)$ |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2739 |
we have |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2740 |
\[ |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2741 |
z_Y(a_1\bullet a_2\bullet b_1\bullet b_2) = |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2742 |
z_{D\times I}(\psi_1(a_1\ot b_1)\bullet \psi_2(a_2\ot b_2)) . |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2743 |
\] |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2744 |
In other words, the inner product on $Y$ is determined by the inner products on |
027bfdae3098
define compatible familty of non-degenerate IPs
Kevin Walker <kevin@canyon23.net>
parents:
387
diff
changeset
|
2745 |
$Y_1$, $Y_2$ and $D\times I$. |
207 | 2746 |
|
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2747 |
Now we show how to unambiguously identify $\cS(X; c; E)$ and $\cS(X; c; E')$ for any |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2748 |
two choices of $E$ and $E'$. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2749 |
Consider first the case where $\bd X$ is decomposed as three $n$-balls $A$, $B$ and $C$, |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2750 |
with $E = \bd(A\cup B)$ and $E' = \bd A$. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2751 |
We must provide an isomorphism between $\cS(X; c; E) = \hom(\cS(C), \cS(A\cup B))$ |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2752 |
and $\cS(X; c; E') = \hom(\cS(C\cup \ol{B}), \cS(A))$. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2753 |
Let $D = B\cap A$. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2754 |
Then as above we can construct a map |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2755 |
\[ |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2756 |
\psi: \cS(B)\ot\cS(\ol{B}) \to \cS(D\times I) . |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2757 |
\] |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2758 |
Given $f\in \hom(\cS(C), \cS(A\cup B))$ we define $f'\in \hom(\cS(C\cup \ol{B}), \cS(A))$ |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2759 |
to be the composition |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2760 |
\[ |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2761 |
\cS(C\cup \ol{B}) \stackrel{f\ot\id}{\longrightarrow} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2762 |
\cS(A\cup B\cup \ol{B}) \stackrel{\id\ot\psi}{\longrightarrow} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2763 |
\cS(A\cup(D\times I)) \stackrel{\cong}{\longrightarrow} \cS(A) . |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2764 |
\] |
393 | 2765 |
(See Figure \ref{jun23b}.) |
2766 |
\begin{figure}[t] |
|
443 | 2767 |
$$ |
2768 |
\begin{tikzpicture}[baseline,line width = 1pt,x=1.5cm,y=1.5cm] |
|
2769 |
\draw (0,0) node(R) {} |
|
2770 |
-- (0.75,0) node[below] {$\bar{B}$} |
|
2771 |
--(1.5,0) node[circle,fill=black,inner sep=2pt] {} |
|
2772 |
arc (0:80:1.5) node[above] {$D \times I$} |
|
2773 |
arc (80:180:1.5); |
|
2774 |
\foreach \r in {0.3, 0.6, 0.9, 1.2} { |
|
2775 |
\draw[blue!50, line width = 0.5pt] (\r,0) arc (0:180:\r); |
|
2776 |
} |
|
2777 |
\draw[fill=white] |
|
2778 |
(R) node[circle,fill=black,inner sep=2pt] {} |
|
2779 |
arc (45:65:3) node[below] {$B$} |
|
2780 |
arc (65:90:3) node[below] {$A$} |
|
2781 |
arc (90:135:3) node[circle,fill=black,inner sep=2pt] {} |
|
2782 |
arc (-135:-90:3) node[below] {$C$} |
|
2783 |
arc (-90:-45:3); |
|
2784 |
\draw[fill] (150:1.5) circle (2pt) node[above=4pt] {$D$}; |
|
547
fbad527790c1
minor: futzing with font size in 2 figs
Kevin Walker <kevin@canyon23.net>
parents:
546
diff
changeset
|
2785 |
\node[green!50!brown] at (-2,0) {\scalebox{1.4}{$\uparrow f$}}; |
fbad527790c1
minor: futzing with font size in 2 figs
Kevin Walker <kevin@canyon23.net>
parents:
546
diff
changeset
|
2786 |
\node[green!50!brown] at (0.2,0.8) {\scalebox{1.4}{$\uparrow \psi$}}; |
443 | 2787 |
\end{tikzpicture} |
2788 |
$$ |
|
393 | 2789 |
\caption{Moving $B$ from top to bottom} |
2790 |
\label{jun23b} |
|
2791 |
\end{figure} |
|
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2792 |
Let $D' = B\cap C$. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2793 |
Using the inner products there is an adjoint map |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2794 |
\[ |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2795 |
\psi^\dagger: \cS(D'\times I) \to \cS(\ol{B})\ot\cS(B) . |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2796 |
\] |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2797 |
Given $f'\in \hom(\cS(C\cup \ol{B}), \cS(A))$ we define $f\in \hom(\cS(C), \cS(A\cup B))$ |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2798 |
to be the composition |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2799 |
\[ |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2800 |
\cS(C) \stackrel{\cong}{\longrightarrow} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2801 |
\cS(C\cup(D'\times I)) \stackrel{\id\ot\psi^\dagger}{\longrightarrow} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2802 |
\cS(C\cup \ol{B}\cup B) \stackrel{f'\ot\id}{\longrightarrow} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2803 |
\cS(A\cup B) . |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2804 |
\] |
393 | 2805 |
(See Figure \ref{jun23c}.) |
2806 |
\begin{figure}[t] |
|
2807 |
\begin{equation*} |
|
443 | 2808 |
\begin{tikzpicture}[baseline,line width = 1pt,x=1.5cm,y=-1.5cm] |
2809 |
\draw (0,0) node(R) {} |
|
2810 |
-- (0.75,0) node[above] {$B$} |
|
2811 |
--(1.5,0) node[circle,fill=black,inner sep=2pt] {} |
|
2812 |
arc (0:80:1.5) node[below] {$D' \times I$} |
|
2813 |
arc (80:180:1.5); |
|
2814 |
\foreach \r in {0.3, 0.6, 0.9, 1.2} { |
|
2815 |
\draw[blue!50, line width = 0.5pt] (\r,0) arc (0:180:\r); |
|
2816 |
} |
|
2817 |
\draw[fill=white] |
|
2818 |
(R) node[circle,fill=black,inner sep=2pt] {} |
|
2819 |
arc (45:65:3) node[above] {$\bar{B}$} |
|
2820 |
arc (65:90:3) node[below] {$C$} |
|
2821 |
arc (90:135:3) node[circle,fill=black,inner sep=2pt] {} |
|
2822 |
arc (-135:-90:3) node[below] {$A$} |
|
2823 |
arc (-90:-45:3); |
|
2824 |
\draw[fill] (150:1.5) circle (2pt) node[below=4pt] {$D'$}; |
|
547
fbad527790c1
minor: futzing with font size in 2 figs
Kevin Walker <kevin@canyon23.net>
parents:
546
diff
changeset
|
2825 |
\node[green!50!brown] at (-2,0) {\scalebox{1.4}{$f'\uparrow $}}; |
fbad527790c1
minor: futzing with font size in 2 figs
Kevin Walker <kevin@canyon23.net>
parents:
546
diff
changeset
|
2826 |
\node[green!50!brown] at (0.2,0.8) {\scalebox{1.4}{$\psi^\dagger \uparrow $}}; |
443 | 2827 |
\end{tikzpicture} |
393 | 2828 |
\end{equation*} |
2829 |
\caption{Moving $B$ from bottom to top} |
|
2830 |
\label{jun23c} |
|
2831 |
\end{figure} |
|
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2832 |
It is not hard too show that the above two maps are mutually inverse. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2833 |
|
559 | 2834 |
\begin{lem} \label{equator-lemma} |
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2835 |
Any two choices of $E$ and $E'$ are related by a series of modifications as above. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2836 |
\end{lem} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2837 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2838 |
\begin{proof} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2839 |
(Sketch) |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2840 |
$E$ and $E'$ are isotopic, and any isotopy is |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2841 |
homotopic to a composition of small isotopies which are either |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2842 |
(a) supported away from $E$, or (b) modify $E$ in the simple manner described above. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2843 |
\end{proof} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2844 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2845 |
It follows from the lemma that we can construct an isomorphism |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2846 |
between $\cS(X; c; E)$ and $\cS(X; c; E')$ for any pair $E$, $E'$. |
855 | 2847 |
This construction involves a choice of simple ``moves" (as above) to transform |
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2848 |
$E$ to $E'$. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2849 |
We must now show that the isomorphism does not depend on this choice. |
855 | 2850 |
We will show below that it suffices to check two ``movie moves". |
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2851 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2852 |
The first movie move is to push $E$ across an $n$-ball $B$ as above, then push it back. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2853 |
The result is equivalent to doing nothing. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2854 |
As we remarked above, the isomorphisms corresponding to these two pushes are mutually |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2855 |
inverse, so we have invariance under this movie move. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2856 |
|
439
10f0f68cafb4
mostly (entirely?) ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
435
diff
changeset
|
2857 |
The second movie move replaces two successive pushes in the same direction, |
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2858 |
across $B_1$ and $B_2$, say, with a single push across $B_1\cup B_2$. |
393 | 2859 |
(See Figure \ref{jun23d}.) |
2860 |
\begin{figure}[t] |
|
456
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2861 |
\begin{tikzpicture} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2862 |
\node(L) { |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2863 |
\scalebox{0.5}{ |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2864 |
\begin{tikzpicture}[baseline,line width = 1pt,x=1.5cm,y=1.5cm] |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2865 |
\draw[red] (0.75,0) -- +(2,0); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2866 |
\draw[red] (0,0) node(R) {} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2867 |
-- (0.75,0) node[below] {} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2868 |
--(1.5,0) node[circle,fill=black,inner sep=2pt] {}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2869 |
\draw[fill] (150:1.5) circle (2pt) node[above=4pt] {}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2870 |
\draw (1.5,0) arc (0:149:1.5); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2871 |
\draw[red] |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2872 |
(R) node[circle,fill=black,inner sep=2pt] {} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2873 |
arc (-45:-135:3) node[circle,fill=black,inner sep=2pt] {}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2874 |
\draw[red] (-5.5,0) -- (-4.2,0); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2875 |
\draw (R) arc (45:75:3); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2876 |
\draw (150:1.5) arc (74:135:3); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2877 |
\node at (-2,0) {\scalebox{2.0}{$B_1$}}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2878 |
\node at (0.2,0.8) {\scalebox{2.0}{$B_2$}}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2879 |
\node at (-4,1.2) {\scalebox{2.0}{$A$}}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2880 |
\node at (-4,-1.2) {\scalebox{2.0}{$C$}}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2881 |
\node[red] at (2.53,0.35) {\scalebox{2.0}{$E$}}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2882 |
\end{tikzpicture} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2883 |
} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2884 |
}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2885 |
\node(M) at (5,4) { |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2886 |
\scalebox{0.5}{ |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2887 |
\begin{tikzpicture}[baseline,line width = 1pt,x=1.5cm,y=1.5cm] |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2888 |
\draw[red] (0.75,0) -- +(2,0); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2889 |
\draw[red] (0,0) node(R) {} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2890 |
-- (0.75,0) node[below] {} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2891 |
--(1.5,0) node[circle,fill=black,inner sep=2pt] {}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2892 |
\draw[fill] (150:1.5) circle (2pt) node[above=4pt] {}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2893 |
\draw(1.5,0) arc (0:149:1.5); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2894 |
\draw |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2895 |
(R) node[circle,fill=black,inner sep=2pt] {} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2896 |
arc (-45:-135:3) node[circle,fill=black,inner sep=2pt] {}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2897 |
\draw[red] (-5.5,0) -- (-4.2,0); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2898 |
\draw[red] (R) arc (45:75:3); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2899 |
\draw[red] (150:1.5) arc (74:135:3); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2900 |
\node at (-2,0) {\scalebox{2.0}{$B_1$}}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2901 |
\node at (0.2,0.8) {\scalebox{2.0}{$B_2$}}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2902 |
\node at (-4,1.2) {\scalebox{2.0}{$A$}}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2903 |
\node at (-4,-1.2) {\scalebox{2.0}{$C$}}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2904 |
\node[red] at (2.53,0.35) {\scalebox{2.0}{$E$}}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2905 |
\end{tikzpicture} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2906 |
} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2907 |
}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2908 |
\node(R) at (10,0) { |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2909 |
\scalebox{0.5}{ |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2910 |
\begin{tikzpicture}[baseline,line width = 1pt,x=1.5cm,y=1.5cm] |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2911 |
\draw[red] (0.75,0) -- +(2,0); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2912 |
\draw (0,0) node(R) {} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2913 |
-- (0.75,0) node[below] {} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2914 |
--(1.5,0) node[circle,fill=black,inner sep=2pt] {}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2915 |
\draw[fill] (150:1.5) circle (2pt) node[above=4pt] {}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2916 |
\draw[red] (1.5,0) arc (0:149:1.5); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2917 |
\draw |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2918 |
(R) node[circle,fill=black,inner sep=2pt] {} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2919 |
arc (-45:-135:3) node[circle,fill=black,inner sep=2pt] {}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2920 |
\draw[red] (-5.5,0) -- (-4.2,0); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2921 |
\draw (R) arc (45:75:3); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2922 |
\draw[red] (150:1.5) arc (74:135:3); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2923 |
\node at (-2,0) {\scalebox{2.0}{$B_1$}}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2924 |
\node at (0.2,0.8) {\scalebox{2.0}{$B_2$}}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2925 |
\node at (-4,1.2) {\scalebox{2.0}{$A$}}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2926 |
\node at (-4,-1.2) {\scalebox{2.0}{$C$}}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2927 |
\node[red] at (2.53,0.35) {\scalebox{2.0}{$E$}}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2928 |
\end{tikzpicture} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2929 |
} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2930 |
}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2931 |
\draw[->] (L) to[out=90,in=225] node[sloped, above] {push $B_1$} (M); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2932 |
\draw[->] (M) to[out=-45,in=90] node[sloped, above] {push $B_2$} (R); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2933 |
\draw[->] (L) to[out=-35,in=-145] node[sloped, below] {push $B_1 \cup B_2$} (R); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2934 |
\end{tikzpicture} |
393 | 2935 |
\caption{A movie move} |
2936 |
\label{jun23d} |
|
2937 |
\end{figure} |
|
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2938 |
Invariance under this movie move follows from the compatibility of the inner |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2939 |
product for $B_1\cup B_2$ with the inner products for $B_1$ and $B_2$. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2940 |
|
505
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2941 |
%The third movie move could be called ``locality" or ``disjoint commutativity". |
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2942 |
%\nn{...} |
439
10f0f68cafb4
mostly (entirely?) ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
435
diff
changeset
|
2943 |
|
855 | 2944 |
If $n\ge 2$, these two movie moves suffice: |
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2945 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2946 |
\begin{lem} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2947 |
Assume $n\ge 2$ and fix $E$ and $E'$ as above. |
550 | 2948 |
Then any two sequences of elementary moves connecting $E$ to $E'$ |
505
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2949 |
are related by a sequence of the two movie moves defined above. |
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2950 |
\end{lem} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2951 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2952 |
\begin{proof} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2953 |
(Sketch) |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2954 |
Consider a two parameter family of diffeomorphisms (one parameter family of isotopies) |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2955 |
of $\bd X$. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2956 |
Up to homotopy, |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2957 |
such a family is homotopic to a family which can be decomposed |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2958 |
into small families which are either |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2959 |
(a) supported away from $E$, |
505
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2960 |
(b) have boundaries corresponding to the two movie moves above. |
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2961 |
Finally, observe that the space of $E$'s is simply connected. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2962 |
(This fails for $n=1$.) |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2963 |
\end{proof} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2964 |
|
855 | 2965 |
For $n=1$ we have to check an additional ``global" relation corresponding to |
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2966 |
rotating the 0-sphere $E$ around the 1-sphere $\bd X$. |
529 | 2967 |
But if $n=1$, then we are in the case of ordinary algebroids and bimodules, |
560 | 2968 |
and this is just the well-known ``Frobenius reciprocity" result for bimodules \cite{MR1424954}. |
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2969 |
|
505
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2970 |
\medskip |
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2971 |
|
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2972 |
We have now defined $\cS(X; c)$ for any $n{+}1$-ball $X$ with boundary decoration $c$. |
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2973 |
We must also define, for any homeomorphism $X\to X'$, an action $f: \cS(X; c) \to \cS(X', f(c))$. |
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2974 |
Choosing an equator $E\sub \bd X$ we have |
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2975 |
\[ |
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2976 |
\cS(X; c) \cong \cS(X; c; E) \deq \hom_{\cS(E_c)}(\cS(\bd_-X_c), \cS(\bd_+X_c)) . |
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2977 |
\] |
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2978 |
We define $f: \cS(X; c) \to \cS(X', f(c))$ to be the tautological map |
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2979 |
\[ |
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2980 |
f: \cS(X; c; E) \to \cS(X'; f(c); f(E)) . |
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2981 |
\] |
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2982 |
It is easy to show that this is independent of the choice of $E$. |
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2983 |
Note also that this map depends only on the restriction of $f$ to $\bd X$. |
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2984 |
In particular, if $F: X\to X$ is the identity on $\bd X$ then $f$ acts trivially, as required by |
552 | 2985 |
Axiom \ref{axiom:extended-isotopies}. |
505
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2986 |
|
506 | 2987 |
We define product $n{+}1$-morphisms to be identity maps of modules. |
101 | 2988 |
|
506 | 2989 |
To define (binary) composition of $n{+}1$-morphisms, choose the obvious common equator |
2990 |
then compose the module maps. |
|
559 | 2991 |
The proof that this composition rule is associative is similar to the proof of Lemma \ref{equator-lemma}. |
803
a96ffd48ea3d
wrote a little (but not enough) about Morita equivalence; out of time, will finish later
Kevin Walker <kevin@canyon23.net>
parents:
802
diff
changeset
|
2992 |
|
a96ffd48ea3d
wrote a little (but not enough) about Morita equivalence; out of time, will finish later
Kevin Walker <kevin@canyon23.net>
parents:
802
diff
changeset
|
2993 |
\medskip |
a96ffd48ea3d
wrote a little (but not enough) about Morita equivalence; out of time, will finish later
Kevin Walker <kevin@canyon23.net>
parents:
802
diff
changeset
|
2994 |
|
865
7abe7642265e
relentless adding 'disk-like' everywhere it could possibly go
Scott Morrison <scott@tqft.net>
parents:
861
diff
changeset
|
2995 |
We end this subsection with some remarks about Morita equivalence of disk-like $n$-categories. |
806 | 2996 |
Recall that two 1-categories $\cC$ and $\cD$ are Morita equivalent if and only if they are equivalent |
811
858b80dfa05c
intertwinor -> intertwiner: http://www.googlefight.com/index.php?lang=en_GB\&word1=intertwiner\&word2=intertwinor
Scott Morrison <scott@tqft.net>
parents:
810
diff
changeset
|
2997 |
objects in the 2-category of (linear) 1-categories, bimodules, and intertwiners. |
865
7abe7642265e
relentless adding 'disk-like' everywhere it could possibly go
Scott Morrison <scott@tqft.net>
parents:
861
diff
changeset
|
2998 |
Similarly, we define two disk-like $n$-categories to be Morita equivalent if they are equivalent objects in the |
803
a96ffd48ea3d
wrote a little (but not enough) about Morita equivalence; out of time, will finish later
Kevin Walker <kevin@canyon23.net>
parents:
802
diff
changeset
|
2999 |
$n{+}1$-category of sphere modules. |
a96ffd48ea3d
wrote a little (but not enough) about Morita equivalence; out of time, will finish later
Kevin Walker <kevin@canyon23.net>
parents:
802
diff
changeset
|
3000 |
|
865
7abe7642265e
relentless adding 'disk-like' everywhere it could possibly go
Scott Morrison <scott@tqft.net>
parents:
861
diff
changeset
|
3001 |
Because of the strong duality enjoyed by disk-like $n$-categories, the data for such an equivalence lives only in |
806 | 3002 |
dimensions 1 and $n+1$ (the middle dimensions come along for free). |
3003 |
The $n{+}1$-dimensional part of the data must be invertible and satisfy |
|
3004 |
identities corresponding to Morse cancellations in $n$-manifolds. |
|
804 | 3005 |
We will treat this in detail for the $n=2$ case; the case for general $n$ is very similar. |
3006 |
||
865
7abe7642265e
relentless adding 'disk-like' everywhere it could possibly go
Scott Morrison <scott@tqft.net>
parents:
861
diff
changeset
|
3007 |
Let $\cC$ and $\cD$ be (unoriented) disk-like 2-categories. |
804 | 3008 |
Let $\cS$ denote the 3-category of 2-category sphere modules. |
806 | 3009 |
The 1-dimensional part of the data for a Morita equivalence between $\cC$ and $\cD$ is a 0-sphere module $\cM = {}_\cC\cM_\cD$ |
3010 |
(categorified bimodule) connecting $\cC$ and $\cD$. |
|
804 | 3011 |
Because of the full unoriented symmetry, this can also be thought of as a |
806 | 3012 |
0-sphere module ${}_\cD\cM_\cC$ connecting $\cD$ and $\cC$. |
804 | 3013 |
|
806 | 3014 |
We want $\cM$ to be an equivalence, so we need 2-morphisms in $\cS$ |
3015 |
between ${}_\cC\cM_\cD \otimes_\cD {}_\cD\cM_\cC$ and the identity 0-sphere module ${}_\cC\cC_\cC$, and similarly |
|
3016 |
with the roles of $\cC$ and $\cD$ reversed. |
|
804 | 3017 |
These 2-morphisms come for free, in the sense of not requiring additional data, since we can take them to be the labeled |
807 | 3018 |
cell complexes (cups and caps) in $B^2$ shown in Figure \ref{morita-fig-1}. |
3019 |
\begin{figure}[t] |
|
3020 |
$$\mathfig{.65}{tempkw/morita1}$$ |
|
812
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3021 |
|
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3022 |
|
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3023 |
$$ |
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3024 |
\begin{tikzpicture} |
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3025 |
\node(L) at (0,0) {\tikz{ |
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3026 |
\draw[orange] (0,0) -- node[below] {$\cC$} (1,0); |
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3027 |
\draw[blue] (1,0) -- node[below] {$\cD$} (2,0); |
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3028 |
\draw[orange] (2,0) -- node[below] {$\cC$} (3,0); |
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3029 |
\node[purple, fill, circle, inner sep=2pt, label=$\cM$] at (1,0) {}; |
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3030 |
\node[purple, fill, circle, inner sep=2pt, label=$\cM$] at (2,0) {}; |
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3031 |
}}; |
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3032 |
|
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3033 |
\node(R) at (6,0) {\tikz{ |
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3034 |
\draw[orange] (0,0) -- node[below] {$\cC$} (3,0); |
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3035 |
\node[label={\phantom{$\cM$}}] at (1.5,0) {}; |
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3036 |
}}; |
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3037 |
|
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3038 |
\node at (-1,-1.5) { $\leftidx{_\cC}{(\cM \tensor_\cD \cM)}{_\cC}$ }; |
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3039 |
\node at (7,-1.5) { $\leftidx{_\cC}{\cC}{_\cC}$ }; |
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3040 |
|
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3041 |
\draw[->] (L) to[out=35, in=145] node[below] {$w$} node[above] { \tikz{ |
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3042 |
\draw (0,0) circle (16pt); |
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3043 |
}}(R); |
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3044 |
|
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3045 |
\draw[->] (R) to[out=-145, in=-35] node[above] {$x$} node[below] { \tikz{ |
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3046 |
\draw (0,0) circle (16pt); |
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3047 |
}}(L); |
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3048 |
|
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3049 |
|
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3050 |
\end{tikzpicture} |
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3051 |
$$ |
14d12dff8268
starting on a Morita figure
Scott Morrison <scott@tqft.net>
parents:
811
diff
changeset
|
3052 |
|
807 | 3053 |
\caption{Cups and caps for free}\label{morita-fig-1} |
3054 |
\end{figure} |
|
3055 |
||
804 | 3056 |
|
3057 |
We want the 2-morphisms from the previous paragraph to be equivalences, so we need 3-morphisms |
|
3058 |
between various compositions of these 2-morphisms and various identity 2-morphisms. |
|
811
858b80dfa05c
intertwinor -> intertwiner: http://www.googlefight.com/index.php?lang=en_GB\&word1=intertwiner\&word2=intertwinor
Scott Morrison <scott@tqft.net>
parents:
810
diff
changeset
|
3059 |
Recall that the 3-morphisms of $\cS$ are intertwiners between representations of 1-categories associated |
804 | 3060 |
to decorated circles. |
807 | 3061 |
Figure \ref{morita-fig-2} |
3062 |
\begin{figure}[t] |
|
3063 |
$$\mathfig{.55}{tempkw/morita2}$$ |
|
811
858b80dfa05c
intertwinor -> intertwiner: http://www.googlefight.com/index.php?lang=en_GB\&word1=intertwiner\&word2=intertwinor
Scott Morrison <scott@tqft.net>
parents:
810
diff
changeset
|
3064 |
\caption{intertwiners for a Morita equivalence}\label{morita-fig-2} |
807 | 3065 |
\end{figure} |
811
858b80dfa05c
intertwinor -> intertwiner: http://www.googlefight.com/index.php?lang=en_GB\&word1=intertwiner\&word2=intertwinor
Scott Morrison <scott@tqft.net>
parents:
810
diff
changeset
|
3066 |
shows the intertwiners we need. |
804 | 3067 |
Each decorated 2-ball in that figure determines a representation of the 1-category associated to the decorated circle |
3068 |
on the boundary. |
|
3069 |
This is the 3-dimensional part of the data for the Morita equivalence. |
|
807 | 3070 |
(Note that, by symmetry, the $c$ and $d$ arrows of Figure \ref{morita-fig-2} |
3071 |
are the same (up to rotation), as are the $h$ and $g$ arrows.) |
|
804 | 3072 |
|
806 | 3073 |
In order for these 3-morphisms to be equivalences, |
3074 |
they must be invertible (i.e.\ $a=b\inv$, $c=d\inv$, $e=f\inv$) and in addition |
|
3075 |
they must satisfy identities corresponding to Morse cancellations on 2-manifolds. |
|
807 | 3076 |
These are illustrated in Figure \ref{morita-fig-3}. |
3077 |
\begin{figure}[t] |
|
3078 |
$$\mathfig{.65}{tempkw/morita3}$$ |
|
811
858b80dfa05c
intertwinor -> intertwiner: http://www.googlefight.com/index.php?lang=en_GB\&word1=intertwiner\&word2=intertwinor
Scott Morrison <scott@tqft.net>
parents:
810
diff
changeset
|
3079 |
\caption{Identities for intertwiners}\label{morita-fig-3} |
807 | 3080 |
\end{figure} |
811
858b80dfa05c
intertwinor -> intertwiner: http://www.googlefight.com/index.php?lang=en_GB\&word1=intertwiner\&word2=intertwinor
Scott Morrison <scott@tqft.net>
parents:
810
diff
changeset
|
3081 |
Each line shows a composition of two intertwiners which we require to be equal to the identity intertwiner. |
817 | 3082 |
The modules corresponding leftmost and rightmost disks in the figure can be identified via the obvious isotopy. |
804 | 3083 |
|
806 | 3084 |
For general $n$, we start with an $n$-category 0-sphere module $\cM$ which is the data for the 1-dimensional |
804 | 3085 |
part of the Morita equivalence. |
3086 |
For $2\le k \le n$, the $k$-dimensional parts of the Morita equivalence are various decorated $k$-balls with submanifolds |
|
806 | 3087 |
labeled by $\cC$, $\cD$ and $\cM$; no additional data is needed for these parts. |
811
858b80dfa05c
intertwinor -> intertwiner: http://www.googlefight.com/index.php?lang=en_GB\&word1=intertwiner\&word2=intertwinor
Scott Morrison <scott@tqft.net>
parents:
810
diff
changeset
|
3088 |
The $n{+}1$-dimensional part of the equivalence is given by certain intertwiners, and these intertwiners must |
806 | 3089 |
be invertible and satisfy |
3090 |
identities corresponding to Morse cancellations in $n$-manifolds. |
|
804 | 3091 |
|
806 | 3092 |
\noop{ |
3093 |
One way of thinking of these conditions is as follows. |
|
3094 |
Given a decorated $n{+}1$-manifold, with a codimension 1 submanifold labeled by $\cM$ and |
|
3095 |
codimension 0 submanifolds labeled by $\cC$ and $\cD$, we can make any local modification we like without |
|
3096 |
changing |
|
3097 |
} |
|
804 | 3098 |
|
806 | 3099 |
If $\cC$ and $\cD$ are Morita equivalent $n$-categories, then it is easy to show that for any $n-j$-manifold |
3100 |
$Y$ the $j$-categories $\cC(Y)$ and $\cD(Y)$ are Morita equivalent. |
|
3101 |
When $j=0$ this means that the TQFT Hilbert spaces $\cC(Y)$ and $\cD(Y)$ are isomorphic |
|
3102 |
(if we are enriching over vector spaces). |
|
804 | 3103 |
|
3104 |
||
3105 |
||
3106 |
||
803
a96ffd48ea3d
wrote a little (but not enough) about Morita equivalence; out of time, will finish later
Kevin Walker <kevin@canyon23.net>
parents:
802
diff
changeset
|
3107 |
\noop{ % the following doesn't work; need 2^(k+1) different N's, not 2*(k+1) |
806 | 3108 |
More specifically, the 1-dimensional part of the data is a 0-sphere module $\cM = {}_\cCM_\cD$ |
3109 |
(categorified bimodule) connecting $\cC$ and $\cD$. |
|
3110 |
From $\cM$ we can construct various $k$-sphere modules $N^k_{j,E}$ for $0 \le k \le n$, $0\le j \le k$, and $E = \cC$ or $\cD$. |
|
803
a96ffd48ea3d
wrote a little (but not enough) about Morita equivalence; out of time, will finish later
Kevin Walker <kevin@canyon23.net>
parents:
802
diff
changeset
|
3111 |
$N^k_{j,E}$ can be thought of as the graph of an index $j$ Morse function on the $k$-ball $B^k$ |
a96ffd48ea3d
wrote a little (but not enough) about Morita equivalence; out of time, will finish later
Kevin Walker <kevin@canyon23.net>
parents:
802
diff
changeset
|
3112 |
(so the graph lives in $B^k\times I = B^{k+1}$). |
a96ffd48ea3d
wrote a little (but not enough) about Morita equivalence; out of time, will finish later
Kevin Walker <kevin@canyon23.net>
parents:
802
diff
changeset
|
3113 |
The positive side of the graph is labeled by $E$, the negative side by $E'$ |
806 | 3114 |
(where $\cC' = \cD$ and $\cD' = \cC$), and the codimension-1 |
3115 |
submanifold separating the positive and negative regions is labeled by $\cM$. |
|
803
a96ffd48ea3d
wrote a little (but not enough) about Morita equivalence; out of time, will finish later
Kevin Walker <kevin@canyon23.net>
parents:
802
diff
changeset
|
3116 |
We think of $N^k_{j,E}$ as a $k{+}1$-morphism connecting |
a96ffd48ea3d
wrote a little (but not enough) about Morita equivalence; out of time, will finish later
Kevin Walker <kevin@canyon23.net>
parents:
802
diff
changeset
|
3117 |
We plan on treating this in more detail in a future paper. |
a96ffd48ea3d
wrote a little (but not enough) about Morita equivalence; out of time, will finish later
Kevin Walker <kevin@canyon23.net>
parents:
802
diff
changeset
|
3118 |
\nn{should add a few more details} |
804 | 3119 |
} |
803
a96ffd48ea3d
wrote a little (but not enough) about Morita equivalence; out of time, will finish later
Kevin Walker <kevin@canyon23.net>
parents:
802
diff
changeset
|
3120 |