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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The definitions presented below tie the categories more closely to the topology |
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and avoid combinatorial questions about, for example, the minimal sufficient |
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collections 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), |
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satisfy our axioms. |
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For examples of a more purely algebraic origin, 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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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 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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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}, \ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product} and \ref{axiom:extended-isotopies}; for an $A_\infty$ $n$-category, we replace Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}. |
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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 $f:\cC(X)\to \cC(Y)$. |
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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 topological or 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 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 with a section 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 fully 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 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 some $k$ 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 at lower levels. |
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%In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point. |
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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 $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
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These maps, for various $X$, comprise a natural transformation of functors. |
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\end{axiom} |
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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. |
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Given $c\in\cl{\cC}(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$. |
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Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
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The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
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all $c\in \cl{\cC}(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category |
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with sufficient limits and colimits |
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(e.g.\ vector spaces, or modules over some ring, or chain complexes), |
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%\nn{actually, need both disj-union/sum and product/tensor-product; what's the name for this sort of cat?} |
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and all the structure maps of the $n$-category should be compatible with the auxiliary |
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category structure. |
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Note that this auxiliary structure is only in dimension $n$; if $\dim(Y) < n$ then |
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$\cC(Y; c)$ is just a plain set. |
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\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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\medskip |
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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. |
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The following lemma will follow from the colimit construction used to define $\cl{\cC}_{k-1}$ |
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on spheres. |
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|
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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 | 173 |
Let $\cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2)$ denote the fibered product of the |
174 |
two maps $\bd: \cC(B_i)\to \cl{\cC}(E)$. |
|
187 | 175 |
Then we have an injective map |
94 | 176 |
\[ |
402 | 177 |
\gl_E : \cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2) \into \cl{\cC}(S) |
94 | 178 |
\] |
187 | 179 |
which is natural with respect to the actions of homeomorphisms. |
333 | 180 |
(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 | 183 |
|
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\begin{figure}[!ht] \centering |
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\begin{tikzpicture}[%every label/.style={green} |
333 | 186 |
] |
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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) {}; |
186 | 189 |
\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 | 195 |
|
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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 | 199 |
|
333 | 200 |
Let $\cl{\cC}(S)_E$ denote the image of $\gl_E$. |
402 | 201 |
We will refer to elements of $\cl{\cC}(S)_E$ as ``splittable along $E$" or ``transverse to $E$". |
109 | 202 |
|
195 | 203 |
If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ |
333 | 204 |
as above, then we define $\cC(X)_E = \bd^{-1}(\cl{\cC}(\bd X)_E)$. |
195 | 205 |
|
333 | 206 |
We will call the projection $\cl{\cC}(S)_E \to \cC(B_i)$ |
110 | 207 |
a {\it restriction} map and write $\res_{B_i}(a)$ |
333 | 208 |
(or simply $\res(a)$ when there is no ambiguity), for $a\in \cl{\cC}(S)_E$. |
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More generally, we also include under the rubric ``restriction map" |
195 | 210 |
the boundary maps of Axiom \ref{nca-boundary} above, |
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another class of maps introduced after Axiom \ref{nca-assoc} below, as well as any composition |
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of restriction maps. |
195 | 213 |
In particular, we have restriction maps $\cC(X)_E \to \cC(B_i)$ |
214 |
($i = 1, 2$, notation from previous paragraph). |
|
215 |
These restriction maps can be thought of as |
|
216 |
domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$. |
|
94 | 217 |
|
218 |
||
219 |
Next we consider composition of morphisms. |
|
220 |
For $n$-categories which lack strong duality, one usually considers |
|
221 |
$k$ different types of composition of $k$-morphisms, each associated to a different direction. |
|
222 |
(For example, vertical and horizontal composition of 2-morphisms.) |
|
223 |
In the presence of strong duality, these $k$ distinct compositions are subsumed into |
|
224 |
one general type of composition which can be in any ``direction". |
|
225 |
||
187 | 226 |
\begin{axiom}[Composition] |
560 | 227 |
\label{axiom:composition} |
187 | 228 |
Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
179 | 229 |
and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
103 | 230 |
Let $E = \bd Y$, which is a $k{-}2$-sphere. |
94 | 231 |
Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
232 |
We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$. |
|
233 |
Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. |
|
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We have a map |
94 | 235 |
\[ |
236 |
\gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E |
|
237 |
\] |
|
238 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
239 |
to the intersection of the boundaries of $B$ and $B_i$. |
|
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If $k < n$, |
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or if $k=n$ and we are in the $A_\infty$ case, |
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we require that $\gl_Y$ is injective. |
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(For $k=n$ in the ordinary (non-$A_\infty$) case, see below.) |
187 | 244 |
\end{axiom} |
94 | 245 |
|
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\begin{figure}[!ht] \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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\end{tikzpicture} |
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\caption{From two balls to one ball.}\label{blah5}\end{figure} |
179 | 259 |
|
195 | 260 |
\begin{axiom}[Strict associativity] \label{nca-assoc} |
187 | 261 |
The composition (gluing) maps above are strictly associative. |
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Given any splitting of a ball $B$ into smaller balls |
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$$\bigsqcup B_i \to B,$$ |
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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 | 265 |
\end{axiom} |
102 | 266 |
|
179 | 267 |
\begin{figure}[!ht] |
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$$\mathfig{.65}{ncat/strict-associativity}$$ |
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\caption{An example of strict associativity.}\label{blah6}\end{figure} |
179 | 270 |
|
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We'll use the notation $a\bullet b$ for the glued together field $\gl_Y(a, b)$. |
110 | 272 |
In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ |
195 | 273 |
a restriction map (one of many types of map so called) and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$. |
274 |
%Compositions of boundary and restriction maps will also be called restriction maps. |
|
275 |
%For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a |
|
276 |
%restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$. |
|
110 | 277 |
|
192 | 278 |
We will write $\cC(B)_Y$ for the image of $\gl_Y$ in $\cC(B)$. |
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We will call elements of $\cC(B)_Y$ morphisms which are |
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``splittable along $Y$'' or ``transverse to $Y$''. |
192 | 281 |
We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$. |
109 | 282 |
|
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More generally, let $\alpha$ be a splitting of $X$ into smaller balls. |
193 | 284 |
Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from |
285 |
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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287 |
In situations where the splitting is notationally anonymous, we will write |
193 | 288 |
$\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to) |
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the unnamed splitting. |
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290 |
If $\beta$ is a ball decomposition of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cl{\cC}(\bd X)_\beta)$; |
193 | 291 |
this can also be denoted $\cC(X)\spl$ if the context contains an anonymous |
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292 |
decomposition of $\bd X$ and no competing splitting of $X$. |
192 | 293 |
|
294 |
The above two composition axioms are equivalent to the following one, |
|
102 | 295 |
which we state in slightly vague form. |
296 |
||
297 |
\xxpar{Multi-composition:} |
|
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{Given any splitting $B_1 \sqcup \cdots \sqcup B_m \to B$ of a $k$-ball |
102 | 299 |
into small $k$-balls, there is a |
300 |
map from an appropriate subset (like a fibered product) |
|
193 | 301 |
of $\cC(B_1)\spl\times\cdots\times\cC(B_m)\spl$ to $\cC(B)\spl$, |
95 | 302 |
and these various $m$-fold composition maps satisfy an |
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303 |
operad-type strict associativity condition (Figure \ref{fig:operad-composition}).} |
179 | 304 |
|
305 |
\begin{figure}[!ht] |
|
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$$\mathfig{.8}{ncat/operad-composition}$$ |
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\caption{Operad composition and associativity}\label{fig:operad-composition}\end{figure} |
95 | 308 |
|
309 |
The next axiom is related to identity morphisms, though that might not be immediately obvious. |
|
310 |
||
343 | 311 |
\begin{axiom}[Product (identity) morphisms, preliminary version] |
312 |
For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$, |
|
313 |
usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. |
|
314 |
These maps must satisfy the following conditions. |
|
315 |
\begin{enumerate} |
|
316 |
\item |
|
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If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are homeomorphisms such that the diagram |
343 | 318 |
\[ \xymatrix{ |
319 |
X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\ |
|
320 |
X \ar[r]^{f} & X' |
|
321 |
} \] |
|
322 |
commutes, then we have |
|
323 |
\[ |
|
324 |
\tilde{f}(a\times D) = f(a)\times D' . |
|
325 |
\] |
|
326 |
\item |
|
327 |
Product morphisms are compatible with gluing (composition) in both factors: |
|
328 |
\[ |
|
329 |
(a'\times D)\bullet(a''\times D) = (a'\bullet a'')\times D |
|
330 |
\] |
|
331 |
and |
|
332 |
\[ |
|
333 |
(a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') . |
|
334 |
\] |
|
335 |
\item |
|
336 |
Product morphisms are associative: |
|
337 |
\[ |
|
338 |
(a\times D)\times D' = a\times (D\times D') . |
|
339 |
\] |
|
340 |
(Here we are implicitly using functoriality and the obvious homeomorphism |
|
341 |
$(X\times D)\times D' \to X\times(D\times D')$.) |
|
342 |
\item |
|
343 |
Product morphisms are compatible with restriction: |
|
344 |
\[ |
|
345 |
\res_{X\times E}(a\times D) = a\times E |
|
346 |
\] |
|
347 |
for $E\sub \bd D$ and $a\in \cC(X)$. |
|
348 |
\end{enumerate} |
|
349 |
\end{axiom} |
|
350 |
||
351 |
We will need to strengthen the above preliminary version of the axiom to allow |
|
352 |
for products which are ``pinched" in various ways along their boundary. |
|
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(See Figure \ref{pinched_prods}.) |
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\begin{figure}[t] |
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$$ |
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\begin{tikzpicture}[baseline=0] |
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\begin{scope} |
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\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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\foreach \x in {0, 0.5, ..., 6} { |
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\draw[green!50!brown] (\x,-2) -- (\x,2); |
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362 |
} |
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363 |
\end{scope} |
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364 |
\draw[blue,line width=1.5pt] (0,-3) -- (5.66,-3); |
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365 |
\draw[->,red,line width=2pt] (2.83,-1.5) -- (2.83,-2.5); |
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366 |
\end{tikzpicture} |
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367 |
\qquad \qquad |
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368 |
\begin{tikzpicture}[baseline=-0.15cm] |
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369 |
\begin{scope} |
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370 |
\path[clip] (0,1) arc (90:135:8 and 4) arc (-135:-90:8 and 4) -- cycle; |
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371 |
\draw[blue,line width=2pt] (0,1) arc (90:135:8 and 4) arc (-135:-90:8 and 4) -- cycle; |
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372 |
\foreach \x in {-6, -5.5, ..., 0} { |
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373 |
\draw[green!50!brown] (\x,-2) -- (\x,2); |
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374 |
} |
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375 |
\end{scope} |
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376 |
\draw[blue,line width=1.5pt] (-5.66,-3.15) -- (0,-3.15); |
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377 |
\draw[->,red,line width=2pt] (-2.83,-1.5) -- (-2.83,-2.5); |
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378 |
\end{tikzpicture} |
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379 |
$$ |
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380 |
\caption{Examples of pinched products}\label{pinched_prods} |
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381 |
\end{figure} |
415
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382 |
(The need for a strengthened version will become apparent in Appendix \ref{sec:comparing-defs} |
685
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383 |
where we construct a traditional category from a disk-like category.) |
343 | 384 |
Define a {\it pinched product} to be a map |
385 |
\[ |
|
386 |
\pi: E\to X |
|
387 |
\] |
|
344 | 388 |
such that $E$ is a $k{+}m$-ball, $X$ is a $k$-ball ($m\ge 1$), and $\pi$ is locally modeled |
343 | 389 |
on a standard iterated degeneracy map |
390 |
\[ |
|
344 | 391 |
d: \Delta^{k+m}\to\Delta^k . |
343 | 392 |
\] |
393 |
(We thank Kevin Costello for suggesting this approach.) |
|
394 |
||
344 | 395 |
Note that for each interior point $x\in X$, $\pi\inv(x)$ is an $m$-ball, |
494
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|
396 |
and for each boundary point $x\in\bd X$, $\pi\inv(x)$ is a ball of dimension |
344 | 397 |
$l \le m$, with $l$ depending on $x$. |
343 | 398 |
It is easy to see that a composition of pinched products is again a pinched product. |
399 |
A {\it sub pinched product} is a sub-$m$-ball $E'\sub E$ such that the restriction |
|
400 |
$\pi:E'\to \pi(E')$ is again a pinched product. |
|
401 |
A {union} of pinched products is a decomposition $E = \cup_i E_i$ |
|
402 |
such that each $E_i\sub E$ is a sub pinched product. |
|
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403 |
(See Figure \ref{pinched_prod_unions}.) |
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|
404 |
\begin{figure}[t] |
364
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405 |
$$ |
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406 |
\begin{tikzpicture}[baseline=0] |
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407 |
\begin{scope} |
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408 |
\path[clip] (0,0) arc (135:45:4) arc (-45:-135:4); |
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409 |
\draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4); |
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410 |
\draw[blue] (0,0) -- (5.66,0); |
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411 |
\foreach \x in {0, 0.5, ..., 6} { |
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412 |
\draw[green!50!brown] (\x,-2) -- (\x,2); |
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413 |
} |
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414 |
\end{scope} |
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415 |
\end{tikzpicture} |
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416 |
\qquad |
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417 |
\begin{tikzpicture}[baseline=0] |
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418 |
\begin{scope} |
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419 |
\path[clip] (0,-1) rectangle (4,1); |
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420 |
\draw[blue,line width=2pt] (0,-1) rectangle (4,1); |
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421 |
\draw[blue] (0,0) -- (5,0); |
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422 |
\foreach \x in {0, 0.5, ..., 6} { |
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423 |
\draw[green!50!brown] (\x,-2) -- (\x,2); |
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424 |
} |
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425 |
\end{scope} |
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426 |
\end{tikzpicture} |
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427 |
\qquad |
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428 |
\begin{tikzpicture}[baseline=0] |
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429 |
\begin{scope} |
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430 |
\path[clip] (0,0) arc (135:45:4) arc (-45:-135:4); |
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431 |
\draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4); |
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432 |
\draw[blue] (2.83,3) circle (3); |
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433 |
\foreach \x in {0, 0.5, ..., 6} { |
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434 |
\draw[green!50!brown] (\x,-2) -- (\x,2); |
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435 |
} |
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436 |
\end{scope} |
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437 |
\end{tikzpicture} |
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438 |
$$ |
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439 |
$$ |
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440 |
\begin{tikzpicture}[baseline=0] |
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441 |
\begin{scope} |
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442 |
\path[clip] (0,-1) rectangle (4,1); |
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443 |
\draw[blue,line width=2pt] (0,-1) rectangle (4,1); |
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444 |
\draw[blue] (0,-1) -- (4,1); |
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445 |
\foreach \x in {0, 0.5, ..., 6} { |
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446 |
\draw[green!50!brown] (\x,-2) -- (\x,2); |
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447 |
} |
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448 |
\end{scope} |
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449 |
\end{tikzpicture} |
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|
450 |
\qquad |
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451 |
\begin{tikzpicture}[baseline=0] |
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452 |
\begin{scope} |
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453 |
\path[clip] (0,-1) rectangle (5,1); |
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454 |
\draw[blue,line width=2pt] (0,-1) rectangle (5,1); |
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455 |
\draw[blue] (1,-1) .. controls (2,-1) and (3,1) .. (4,1); |
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456 |
\foreach \x in {0, 0.5, ..., 6} { |
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457 |
\draw[green!50!brown] (\x,-2) -- (\x,2); |
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458 |
} |
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|
459 |
\end{scope} |
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|
460 |
\end{tikzpicture} |
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|
461 |
$$ |
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|
462 |
\caption{Five examples of unions of pinched products}\label{pinched_prod_unions} |
352
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|
463 |
\end{figure} |
343 | 464 |
|
465 |
The product axiom will give a map $\pi^*:\cC(X)\to \cC(E)$ for each pinched product |
|
466 |
$\pi:E\to X$. |
|
344 | 467 |
Morphisms in the image of $\pi^*$ will be called product morphisms. |
343 | 468 |
Before stating the axiom, we illustrate it in our two motivating examples of $n$-categories. |
469 |
In the case where $\cC(X) = \{f: X\to T\}$, we define $\pi^*(f) = f\circ\pi$. |
|
344 | 470 |
In the case where $\cC(X)$ is the set of all labeled embedded cell complexes $K$ in $X$, |
471 |
define $\pi^*(K) = \pi\inv(K)$, with each codimension $i$ cell $\pi\inv(c)$ labeled by the |
|
472 |
same (traditional) $i$-morphism as the corresponding codimension $i$ cell $c$. |
|
343 | 473 |
|
474 |
||
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|
475 |
%\addtocounter{axiom}{-1} |
187 | 476 |
\begin{axiom}[Product (identity) morphisms] |
560 | 477 |
\label{axiom:product} |
344 | 478 |
For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
479 |
there is a map $\pi^*:\cC(X)\to \cC(E)$. |
|
340
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|
480 |
These maps must satisfy the following conditions. |
191
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|
481 |
\begin{enumerate} |
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|
482 |
\item |
344 | 483 |
If $\pi:E\to X$ and $\pi':E'\to X'$ are pinched products, and |
484 |
if $f:X\to X'$ and $\tilde{f}:E \to E'$ are maps such that the diagram |
|
95 | 485 |
\[ \xymatrix{ |
344 | 486 |
E \ar[r]^{\tilde{f}} \ar[d]_{\pi} & E' \ar[d]^{\pi'} \\ |
95 | 487 |
X \ar[r]^{f} & X' |
488 |
} \] |
|
109 | 489 |
commutes, then we have |
490 |
\[ |
|
344 | 491 |
\pi'^*\circ f = \tilde{f}\circ \pi^*. |
109 | 492 |
\] |
191
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|
493 |
\item |
344 | 494 |
Product morphisms are compatible with gluing (composition). |
495 |
Let $\pi:E\to X$, $\pi_1:E_1\to X_1$, and $\pi_2:E_2\to X_2$ |
|
496 |
be pinched products with $E = E_1\cup E_2$. |
|
497 |
Let $a\in \cC(X)$, and let $a_i$ denote the restriction of $a$ to $X_i\sub X$. |
|
498 |
Then |
|
109 | 499 |
\[ |
344 | 500 |
\pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) . |
109 | 501 |
\] |
191
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|
502 |
\item |
344 | 503 |
Product morphisms are associative. |
423 | 504 |
If $\pi:E\to X$ and $\rho:D\to E$ are pinched products then |
109 | 505 |
\[ |
344 | 506 |
\rho^*\circ\pi^* = (\pi\circ\rho)^* . |
109 | 507 |
\] |
191
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|
508 |
\item |
344 | 509 |
Product morphisms are compatible with restriction. |
510 |
If we have a commutative diagram |
|
511 |
\[ \xymatrix{ |
|
512 |
D \ar@{^(->}[r] \ar[d]_{\rho} & E \ar[d]^{\pi} \\ |
|
513 |
Y \ar@{^(->}[r] & X |
|
514 |
} \] |
|
515 |
such that $\rho$ and $\pi$ are pinched products, then |
|
110 | 516 |
\[ |
344 | 517 |
\res_D\circ\pi^* = \rho^*\circ\res_Y . |
110 | 518 |
\] |
191
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|
519 |
\end{enumerate} |
187 | 520 |
\end{axiom} |
95 | 521 |
|
343 | 522 |
|
523 |
\medskip |
|
128 | 524 |
|
95 | 525 |
All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |
526 |
The last axiom (below), concerning actions of |
|
527 |
homeomorphisms in the top dimension $n$, distinguishes the two cases. |
|
528 |
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529 |
We start with the ordinary $n$-category case. |
95 | 530 |
|
420 | 531 |
\begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$] |
187 | 532 |
Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
95 | 533 |
to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
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534 |
Then $f$ acts trivially on $\cC(X)$; that is $f(a) = a$ for all $a\in \cC(X)$. |
267 | 535 |
\end{axiom} |
96 | 536 |
|
174 | 537 |
This axiom needs to be strengthened to force product morphisms to act as the identity. |
103 | 538 |
Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. |
96 | 539 |
Let $J$ be a 1-ball (interval). |
540 |
We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$. |
|
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(Here we use $Y\times J$ with boundary entirely pinched.) |
96 | 542 |
We define a map |
543 |
\begin{eqnarray*} |
|
544 |
\psi_{Y,J}: \cC(X) &\to& \cC(X) \\ |
|
545 |
a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) . |
|
546 |
\end{eqnarray*} |
|
142 | 547 |
(See Figure \ref{glue-collar}.) |
189 | 548 |
\begin{figure}[!ht] |
549 |
\begin{equation*} |
|
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\begin{tikzpicture} |
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\def\rad{1} |
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\def\srad{0.75} |
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\def\gap{4.5} |
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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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\node[right=1mm] at (0.east) {$a$}; |
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\draw (1-small) circle (\srad); |
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\draw[blue] (1) -- ($(1)+(\rad,0)+(\theta:\srad)$); |
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} |
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\filldraw[fill=white] (1) circle (\rad); |
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\foreach \n in {1,2} { |
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\fill (intersection \n of 1-small and 1) circle (2pt); |
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} |
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\node[below] at (1-small.south) {$a \times J$}; |
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|
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\begin{scope} |
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\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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\draw[blue] ($(2)+(-1,\y) $)-- ($(2)+(1,\y)$); |
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} |
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585 |
\end{scope} |
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586 |
\end{tikzpicture} |
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|
587 |
\end{equation*} |
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|
588 |
\begin{equation*} |
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|
589 |
\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 | 590 |
\end{equation*} |
591 |
||
592 |
\caption{Extended homeomorphism.}\label{glue-collar}\end{figure} |
|
415
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|
593 |
We call a map of this form a {\it collar map}. |
96 | 594 |
It can be thought of as the action of the inverse of |
415
8dedd2914d10
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parents:
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diff
changeset
|
595 |
a map which projects a collar neighborhood of $Y$ onto $Y$, |
8dedd2914d10
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parents:
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diff
changeset
|
596 |
or as the limit of homeomorphisms $X\to X$ which expand a very thin collar of $Y$ |
8dedd2914d10
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parents:
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diff
changeset
|
597 |
to a larger collar. |
8dedd2914d10
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diff
changeset
|
598 |
We call the equivalence relation generated by collar maps and homeomorphisms |
8dedd2914d10
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diff
changeset
|
599 |
isotopic (rel boundary) to the identity {\it extended isotopy}. |
96 | 600 |
|
601 |
The revised axiom is |
|
602 |
||
551
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|
603 |
%\addtocounter{axiom}{-1} |
680
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|
604 |
\begin{axiom}[\textup{\textbf{[ordinary version]}} Extended isotopy invariance in dimension $n$.] |
187 | 605 |
\label{axiom:extended-isotopies} |
606 |
Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
|
415
8dedd2914d10
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diff
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|
607 |
to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
187 | 608 |
Then $f$ acts trivially on $\cC(X)$. |
415
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|
609 |
In addition, collar maps act trivially on $\cC(X)$. |
266
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|
610 |
\end{axiom} |
96 | 611 |
|
97 | 612 |
\smallskip |
613 |
||
614 |
For $A_\infty$ $n$-categories, we replace |
|
615 |
isotopy invariance with the requirement that families of homeomorphisms act. |
|
616 |
For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
|
416 | 617 |
Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and |
618 |
$C_*(\Homeo_\bd(X))$ denote the singular chains on this space. |
|
619 |
||
97 | 620 |
|
551
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|
621 |
%\addtocounter{axiom}{-1} |
420 | 622 |
\begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
560 | 623 |
\label{axiom:families} |
335
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|
624 |
For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
97 | 625 |
\[ |
626 |
C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
|
627 |
\] |
|
475
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|
628 |
These action maps are required to be associative up to homotopy, |
07c18e2abd8f
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|
629 |
%\nn{iterated homotopy?} |
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|
630 |
and also compatible with composition (gluing) in the sense that |
437 | 631 |
a diagram like the one in Theorem \ref{thm:CH} commutes. |
475
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|
632 |
%\nn{repeat diagram here?} |
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|
633 |
%\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} |
679
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|
634 |
On $C_0(\Homeo_\bd(X))\ot \cC(X; c)$ the action should coincide |
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|
635 |
with the one coming from Axiom \ref{axiom:morphisms}. |
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|
636 |
\end{axiom} |
97 | 637 |
|
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|
638 |
We should strengthen the above $A_\infty$ axiom to apply to families of collar maps. |
416 | 639 |
To do this we need to explain how collar maps form a topological space. |
640 |
Roughly, the set of collared $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
|
97 | 641 |
and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
416 | 642 |
Having chains on the space of collar maps act gives rise to coherence maps involving |
643 |
weak identities. |
|
420 | 644 |
We will not pursue this in detail here. |
97 | 645 |
|
687
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|
646 |
A potential variant on the above axiom would be to drop the ``up to homotopy" and require a strictly associative action. (In fact, the alternative construction of the blob complex described in \S \ref{ss:alt-def} gives $n$-categories as in Example \ref{ex:blob-complexes-of-balls} which satisfy this stronger axiom; since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across.) |
679
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|
647 |
|
103 | 648 |
Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
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|
649 |
into a ordinary $n$-category (enriched over graded groups). |
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|
650 |
In a different direction, if we enrich over topological spaces instead of chain complexes, |
97 | 651 |
we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
652 |
instead of $C_*(\Homeo_\bd(X))$. |
|
266
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|
653 |
Taking singular chains converts such a space type $A_\infty$ $n$-category into a chain complex |
97 | 654 |
type $A_\infty$ $n$-category. |
655 |
||
99 | 656 |
\medskip |
97 | 657 |
|
680
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|
658 |
The alert reader will have already noticed that our definition of a (ordinary) $n$-category |
416 | 659 |
is extremely similar to our definition of a system of fields. |
660 |
There are two differences. |
|
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|
661 |
First, for the $n$-category definition we restrict our attention to balls |
99 | 662 |
(and their boundaries), while for fields we consider all manifolds. |
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|
663 |
Second, in category definition we directly impose isotopy |
416 | 664 |
invariance in dimension $n$, while in the fields definition we |
665 |
instead remember a subspace of local relations which contain differences of isotopic fields. |
|
340
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|
666 |
(Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.) |
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|
667 |
Thus a system of fields and local relations $(\cF,U)$ determines an $n$-category $\cC_ {\cF,U}$ simply by restricting our attention to |
329
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668 |
balls and, at level $n$, quotienting out by the local relations: |
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|
669 |
\begin{align*} |
494
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|
670 |
\cC_{\cF,U}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / U(B) & \text{when $k=n$.}\end{cases} |
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|
671 |
\end{align*} |
142 | 672 |
This $n$-category can be thought of as the local part of the fields. |
685
8efbd2730ef9
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|
673 |
Conversely, given a disk-like $n$-category we can construct a system of fields via |
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674 |
a colimit construction; see \S \ref{ss:ncat_fields} below. |
99 | 675 |
|
682
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|
676 |
In the $n$-category axioms above we have intermingled data and properties for expository reasons. |
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|
677 |
Here's a summary of the definition which segregates the data from the properties. |
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678 |
|
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679 |
An $n$-category consists of the following data: |
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680 |
\begin{itemize} |
689
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|
681 |
\item functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms}); |
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|
682 |
\item boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary}); |
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|
683 |
\item ``composition'' or ``gluing'' maps $\gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B_1\cup_Y B_2)_E$ (Axiom \ref{axiom:composition}); |
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|
684 |
\item ``product'' or ``identity'' maps $\pi^*:\cC(X)\to \cC(E)$ for each pinched product $\pi:E\to X$ (Axiom \ref{axiom:product}); |
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|
685 |
\item if enriching in an auxiliary category, additional structure on $\cC_n(X; c)$; |
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|
686 |
\item in the $A_\infty$ case, an action of $C_*(\Homeo_\bd(X))$, and similarly for families of collar maps (Axiom \ref{axiom:families}). |
682
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|
687 |
\end{itemize} |
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|
688 |
The above data must satisfy the following conditions: |
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|
689 |
\begin{itemize} |
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|
690 |
\item The gluing maps are compatible with actions of homeomorphisms and boundary |
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|
691 |
restrictions (Axiom \ref{axiom:composition}). |
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|
692 |
\item For $k<n$ the gluing maps are injective (Axiom \ref{axiom:composition}). |
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|
693 |
\item The gluing maps are strictly associative (Axiom \ref{nca-assoc}). |
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|
694 |
\item The product maps are associative and also compatible with homeomorphism actions, gluing and restriction (Axiom \ref{axiom:product}). |
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695 |
\item If enriching in an auxiliary category, all of the data should be compatible |
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|
696 |
with the auxiliary category structure on $\cC_n(X; c)$. |
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|
697 |
\item For ordinary categories, invariance of $n$-morphisms under extended isotopies (Axiom \ref{axiom:extended-isotopies}). |
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|
698 |
\end{itemize} |
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|
699 |
|
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|
700 |
|
512
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|
701 |
\subsection{Examples of \texorpdfstring{$n$}{n}-categories} |
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702 |
\label{ss:ncat-examples} |
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703 |
|
101 | 704 |
|
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705 |
We now describe several classes of examples of $n$-categories satisfying our axioms. |
418
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706 |
We typically specify only the morphisms; the rest of the data for the category |
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707 |
(restriction maps, gluing, product morphisms, action of homeomorphisms) is usually obvious. |
101 | 708 |
|
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709 |
\begin{example}[Maps to a space] |
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710 |
\rm |
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711 |
\label{ex:maps-to-a-space}% |
425 | 712 |
Let $T$ be a topological space. |
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713 |
We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows. |
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714 |
For $X$ a $k$-ball with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of |
191
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715 |
all continuous maps from $X$ to $T$. |
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716 |
For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo |
196 | 717 |
homotopies fixed on $\bd X$. |
101 | 718 |
(Note that homotopy invariance implies isotopy invariance.) |
719 |
For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
|
720 |
be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. |
|
418
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|
721 |
\end{example} |
313 | 722 |
|
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|
723 |
\noop{ |
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|
724 |
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. |
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725 |
Constructing a system of fields from $\pi_{\leq n}(T)$ recovers that example. |
418
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|
726 |
\nn{shouldn't this go elsewhere? we haven't yet discussed constructing a system of fields from |
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|
727 |
an n-cat} |
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|
728 |
} |
101 | 729 |
|
423 | 730 |
\begin{example}[Maps to a space, with a fiber] \label{ex:maps-with-fiber} |
191
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|
731 |
\rm |
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|
732 |
\label{ex:maps-to-a-space-with-a-fiber}% |
196 | 733 |
We can modify the example above, by fixing a |
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|
734 |
closed $m$-manifold $F$, and defining $\pi^{\times F}_{\leq n}(T)(X) = \Maps(X \times F \to T)$, |
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735 |
otherwise leaving the definition in Example \ref{ex:maps-to-a-space} unchanged. |
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|
736 |
Taking $F$ to be a point recovers the previous case. |
191
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|
737 |
\end{example} |
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738 |
|
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|
739 |
\begin{example}[Linearized, twisted, maps to a space] |
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740 |
\rm |
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|
741 |
\label{ex:linearized-maps-to-a-space}% |
191
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|
742 |
We can linearize Examples \ref{ex:maps-to-a-space} and \ref{ex:maps-to-a-space-with-a-fiber} as follows. |
101 | 743 |
Let $\alpha$ be an $(n{+}m{+}1)$-cocycle on $T$ with values in a ring $R$ |
191
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|
744 |
(have in mind the trivial cocycle). |
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|
745 |
For $X$ of dimension less than $n$ define $\pi^{\alpha, \times F}_{\leq n}(T)(X)$ as before, ignoring $\alpha$. |
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|
746 |
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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|
747 |
the $R$-module of finite linear combinations of continuous maps from $X\times F$ to $T$, |
101 | 748 |
modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy |
191
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|
749 |
$h: X\times F\times I \to T$, then $a = \alpha(h)b$. |
418
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|
750 |
(In order for this to be well-defined we must choose $\alpha$ to be zero on degenerate simplices. |
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|
751 |
Alternatively, we could equip the balls with fundamental classes.) |
190
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|
752 |
\end{example} |
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753 |
|
425 | 754 |
\begin{example}[$n$-categories from TQFTs] |
755 |
\rm |
|
756 |
\label{ex:ncats-from-tqfts}% |
|
757 |
Let $\cF$ be a TQFT in the sense of \S\ref{sec:fields}: an $n$-dimensional |
|
758 |
system of fields (also denoted $\cF$) and local relations. |
|
759 |
Let $W$ be an $n{-}j$-manifold. |
|
760 |
Define the $j$-category $\cF(W)$ as follows. |
|
761 |
If $X$ is a $k$-ball with $k<j$, let $\cF(W)(X) \deq \cF(W\times X)$. |
|
762 |
If $X$ is a $j$-ball and $c\in \cl{\cF(W)}(\bd X)$, |
|
763 |
let $\cF(W)(X; c) \deq A_\cF(W\times X; c)$. |
|
764 |
\end{example} |
|
765 |
||
766 |
The next example is only intended to be illustrative, as we don't specify |
|
767 |
which definition of a ``traditional $n$-category" we intend. |
|
768 |
Further, most of these definitions don't even have an agreed-upon notion of |
|
769 |
``strong duality", which we assume here. |
|
191
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|
770 |
\begin{example}[Traditional $n$-categories] |
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771 |
\rm |
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772 |
\label{ex:traditional-n-categories} |
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|
773 |
Given a ``traditional $n$-category with strong duality" $C$ |
310
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|
774 |
define $\cC(X)$, for $X$ a $k$-ball with $k < n$, |
346
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|
775 |
to be the set of all $C$-labeled embedded cell complexes of $X$ (c.f. \S \ref{sec:fields}). |
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|
776 |
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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|
777 |
combinations of $C$-labeled embedded cell complexes of $X$ |
191
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|
778 |
modulo the kernel of the evaluation map. |
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|
779 |
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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780 |
with each cell labelled according to the corresponding cell for $a$. |
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|
781 |
(These two cells have the same codimension.) |
191
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782 |
More generally, start with an $n{+}m$-category $C$ and a closed $m$-manifold $F$. |
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783 |
Define $\cC(X)$, for $\dim(X) < n$, |
346
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|
784 |
to be the set of all $C$-labeled embedded cell complexes of $X\times F$. |
191
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785 |
Define $\cC(X; c)$, for $X$ an $n$-ball, |
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786 |
to be the dual Hilbert space $A(X\times F; c)$. |
426
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|
787 |
(See \S\ref{sec:constructing-a-tqft}.) |
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|
788 |
\end{example} |
313 | 789 |
|
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|
790 |
\noop{ |
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|
791 |
\nn{shouldn't this go elsewhere? we haven't yet discussed constructing a system of fields from |
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|
792 |
an n-cat} |
417
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|
793 |
Recall we described a system of fields and local relations based on a ``traditional $n$-category" |
340
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794 |
$C$ in Example \ref{ex:traditional-n-categories(fields)} above. |
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795 |
\nn{KW: We already refer to \S \ref{sec:fields} above} |
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796 |
Constructing a system of fields from $\cC$ recovers that example. |
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797 |
\todo{Except that it doesn't: pasting diagrams v.s. string diagrams.} |
346
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798 |
\nn{KW: but the above example is all about string diagrams. the only difference is at the top level, |
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799 |
where the quotient is built in. |
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800 |
but (string diagrams)/(relations) is isomorphic to |
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801 |
(pasting diagrams composed of smaller string diagrams)/(relations)} |
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|
802 |
} |
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|
803 |
|
204 | 804 |
|
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|
805 |
\begin{example}[The bordism $n$-category, ordinary version] |
348
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|
806 |
\label{ex:bord-cat} |
191
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|
807 |
\rm |
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808 |
\label{ex:bordism-category} |
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809 |
For a $k$-ball $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional |
191
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|
810 |
submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse |
196 | 811 |
to $\bd X$. |
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812 |
For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes (rel boundary) of such $n$-dimensional submanifolds; |
191
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|
813 |
we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism |
196 | 814 |
$W \to W'$ which restricts to the identity on the boundary. |
191
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|
815 |
\end{example} |
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816 |
|
196 | 817 |
%\nn{the next example might be an unnecessary distraction. consider deleting it.} |
101 | 818 |
|
196 | 819 |
%\begin{example}[Variation on the above examples] |
820 |
%We could allow $F$ to have boundary and specify boundary conditions on $X\times \bd F$, |
|
821 |
%for example product boundary conditions or take the union over all boundary conditions. |
|
822 |
%%\nn{maybe should not emphasize this case, since it's ``better" in some sense |
|
823 |
%%to think of these guys as affording a representation |
|
824 |
%%of the $n{+}1$-category associated to $\bd F$.} |
|
825 |
%\end{example} |
|
101 | 826 |
|
827 |
||
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828 |
%We have two main examples of $A_\infty$ $n$-categories, coming from maps to a target space and from the blob complex. |
101 | 829 |
|
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830 |
\begin{example}[Chains (or space) of maps to a space] |
191
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|
831 |
\rm |
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832 |
\label{ex:chains-of-maps-to-a-space} |
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|
833 |
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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|
834 |
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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|
835 |
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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|
836 |
\[ |
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|
837 |
C_*(\Maps_c(X\times F \to T)), |
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|
838 |
\] |
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|
839 |
where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
101 | 840 |
and $C_*$ denotes singular chains. |
418
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|
841 |
Alternatively, if we take the $n$-morphisms to be simply $\Maps_c(X\times F \to T)$, |
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|
842 |
we get an $A_\infty$ $n$-category enriched over spaces. |
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|
843 |
\end{example} |
101 | 844 |
|
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|
845 |
See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to |
494
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|
846 |
homotopy as the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
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|
847 |
|
279 | 848 |
\begin{example}[Blob complexes of balls (with a fiber)] |
849 |
\rm |
|
850 |
\label{ex:blob-complexes-of-balls} |
|
418
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|
851 |
Fix an $n{-}k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$. |
291 | 852 |
We will define an $A_\infty$ $k$-category $\cC$. |
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|
853 |
When $X$ is a $m$-ball, with $m<k$, define $\cC(X) = \cE(X\times F)$. |
291 | 854 |
When $X$ is an $k$-ball, |
279 | 855 |
define $\cC(X; c) = \bc^\cE_*(X\times F; c)$ |
856 |
where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
|
857 |
\end{example} |
|
101 | 858 |
|
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|
859 |
This example will be used in Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product. |
685
8efbd2730ef9
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|
860 |
Notice that with $F$ a point, the above example is a construction turning an ordinary |
456
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|
861 |
$n$-category $\cC$ into an $A_\infty$ $n$-category. |
417
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|
862 |
We think of this as providing a ``free resolution" |
685
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|
863 |
of the ordinary $n$-category. |
475
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|
864 |
%\nn{say something about cofibrant replacements?} |
340
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|
865 |
In fact, there is also a trivial, but mostly uninteresting, way to do this: |
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|
866 |
we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, |
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|
867 |
and take $\CD{B}$ to act trivially. |
266
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|
868 |
|
685
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|
869 |
Beware that the ``free resolution" of the ordinary $n$-category $\pi_{\leq n}(T)$ |
552 | 870 |
is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. |
340
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|
871 |
It's easy to see that with $n=0$, the corresponding system of fields is just |
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|
872 |
linear combinations of connected components of $T$, and the local relations are trivial. |
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|
873 |
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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|
874 |
|
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|
875 |
\begin{example}[The bordism $n$-category, $A_\infty$ version] |
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|
876 |
\rm |
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|
877 |
\label{ex:bordism-category-ainf} |
348
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|
878 |
As in Example \ref{ex:bord-cat}, for $X$ a $k$-ball, $k<n$, we define $\Bord^{n,\infty}(X)$ |
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|
879 |
to be the set of all $k$-dimensional |
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|
880 |
submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse |
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|
881 |
to $\bd X$. |
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|
882 |
For an $n$-ball $X$ with boundary condition $c$ |
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|
883 |
define $\Bord^{n,\infty}(X; c)$ to be the space of all $k$-dimensional |
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|
884 |
submanifolds $W$ of $X\times \Real^\infty$ such that |
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|
885 |
$W$ coincides with $c$ at $\bd X \times \Real^\infty$. |
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|
886 |
(The topology on this space is induced by ambient isotopy rel boundary. |
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|
887 |
This is homotopy equivalent to a disjoint union of copies $\mathrm{B}\!\Homeo(W')$, where |
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|
888 |
$W'$ runs though representatives of homeomorphism types of such manifolds.) |
309
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|
889 |
\end{example} |
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|
890 |
|
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|
891 |
|
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|
892 |
|
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|
893 |
Let $\cE\cB_n$ be the operad of smooth embeddings of $k$ (little) |
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|
894 |
copies of the standard $n$-ball $B^n$ into another (big) copy of $B^n$. |
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|
895 |
(We require that the interiors of the little balls be disjoint, but their |
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|
896 |
boundaries are allowed to meet. |
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|
897 |
Note in particular that the space for $k=1$ contains a copy of $\Diff(B^n)$, namely |
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|
898 |
the embeddings of a ``little" ball with image all of the big ball $B^n$. |
475
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|
899 |
(But note also that this inclusion is not |
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|
900 |
necessarily a homotopy equivalence.) |
419
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|
901 |
The operad $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad: |
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|
902 |
by shrinking the little balls (precomposing them with dilations), |
346
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|
903 |
we see that both operads are homotopic to the space of $k$ framed points |
401
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|
904 |
in $B^n$. |
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|
905 |
It is easy to see that $n$-fold loop spaces $\Omega^n(T)$ have |
346
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|
906 |
an action of $\cE\cB_n$. |
475
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|
907 |
%\nn{add citation for this operad if we can find one} |
346
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|
908 |
|
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|
909 |
\begin{example}[$E_n$ algebras] |
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|
910 |
\rm |
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|
911 |
\label{ex:e-n-alg} |
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|
912 |
Let $A$ be an $\cE\cB_n$-algebra. |
346
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|
913 |
Note that this implies a $\Diff(B^n)$ action on $A$, |
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|
914 |
since $\cE\cB_n$ contains a copy of $\Diff(B^n)$. |
309
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|
915 |
We will define an $A_\infty$ $n$-category $\cC^A$. |
346
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|
916 |
If $X$ is a ball of dimension $k<n$, define $\cC^A(X)$ to be a point. |
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|
917 |
In other words, the $k$-morphisms are trivial for $k<n$. |
347
14643c4931bc
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|
918 |
If $X$ is an $n$-ball, we define $\cC^A(X)$ via a colimit construction. |
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|
919 |
(Plain colimit, not homotopy colimit.) |
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|
920 |
Let $J$ be the category whose objects are embeddings of a disjoint union of copies of |
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|
921 |
the standard ball $B^n$ into $X$, and who morphisms are given by engulfing some of the |
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|
922 |
embedded balls into a single larger embedded ball. |
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diff
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|
923 |
To each object of $J$ we associate $A^{\times m}$ (where $m$ is the number of balls), and |
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|
924 |
to each morphism of $J$ we associate a morphism coming from the $\cE\cB_n$ action on $A$. |
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|
925 |
Alternatively and more simply, we could define $\cC^A(X)$ to be |
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diff
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|
926 |
$\Diff(B^n\to X)\times A$ modulo the diagonal action of $\Diff(B^n)$. |
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|
927 |
The remaining data for the $A_\infty$ $n$-category |
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|
928 |
--- composition and $\Diff(X\to X')$ action --- |
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|
929 |
also comes from the $\cE\cB_n$ action on $A$. |
528
96ec10a46ee1
minor; resolving a few \nns
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|
930 |
%\nn{should we spell this out?} |
346
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|
931 |
|
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
|
932 |
Conversely, one can show that a disk-like $A_\infty$ $n$-category $\cC$, where the $k$-morphisms |
356
9bbe6eb6fb6c
remark about EB_n-algebras from n-cats
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diff
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|
933 |
$\cC(X)$ are trivial (single point) for $k<n$, gives rise to |
9bbe6eb6fb6c
remark about EB_n-algebras from n-cats
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|
934 |
an $\cE\cB_n$-algebra. |
528
96ec10a46ee1
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diff
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|
935 |
%\nn{The paper is already long; is it worth giving details here?} |
506 | 936 |
|
937 |
If we apply the homotopy colimit construction of the next subsection to this example, |
|
938 |
we get an instance of Lurie's topological chiral homology construction. |
|
191
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diff
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|
939 |
\end{example} |
95 | 940 |
|
108 | 941 |
|
310
ee7be19ee61a
converting sphere axiom to a proposition; still need to make similar changes in module axioms
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diff
changeset
|
942 |
\subsection{From balls to manifolds} |
ee7be19ee61a
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diff
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|
943 |
\label{ss:ncat_fields} \label{ss:ncat-coend} |
552 | 944 |
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
|
945 |
(of either the ordinary or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$. |
552 | 946 |
This extension is a certain colimit, and the arrow in the notation is intended as a reminder of this. |
947 |
||
680
0591d017e698
plain n-cat -> ordinary n-cat
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diff
changeset
|
948 |
In the case of ordinary $n$-categories, this construction factors into a construction of a |
340
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|
949 |
system of fields and local relations, followed by the usual TQFT definition of a |
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|
950 |
vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}. |
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|
951 |
For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. |
680
0591d017e698
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679
diff
changeset
|
952 |
Recall that we can take a ordinary $n$-category $\cC$ and pass to the ``free resolution", |
475
07c18e2abd8f
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diff
changeset
|
953 |
an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls |
07c18e2abd8f
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changeset
|
954 |
(recall Example \ref{ex:blob-complexes-of-balls} above). |
340
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|
955 |
We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant |
475
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changeset
|
956 |
for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the |
552 | 957 |
same as the original blob complex for $M$ with coefficients in $\cC$. |
958 |
||
959 |
Recall that we've already anticipated this construction in the previous section, |
|
960 |
inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls, |
|
961 |
so that we can state the boundary axiom for $\cC$ on $k+1$-balls. |
|
962 |
||
963 |
\medskip |
|
108 | 964 |
|
475
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|
965 |
We will first define the ``decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
340
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|
966 |
An $n$-category $\cC$ provides a functor from this poset to the category of sets, |
419
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|
967 |
and we will define $\cl{\cC}(W)$ as a suitable colimit |
340
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|
968 |
(or homotopy colimit in the $A_\infty$ case) of this functor. |
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|
969 |
We'll later give a more explicit description of this colimit. |
420 | 970 |
In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-balls with boundary data), |
340
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|
971 |
then the resulting colimit is also enriched, that is, the set associated to $W$ splits into subsets according to boundary data, and each of these subsets has the appropriate structure (e.g. a vector space or chain complex). |
108 | 972 |
|
475
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changeset
|
973 |
Recall (Definition \ref{defn:gluing-decomposition}) that a {\it ball decomposition} of $W$ is a |
07c18e2abd8f
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changeset
|
974 |
sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls |
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changeset
|
975 |
$\du_a X_a$. |
07c18e2abd8f
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changeset
|
976 |
Abusing notation, we let $X_a$ denote both the ball (component of $M_0$) and |
07c18e2abd8f
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changeset
|
977 |
its image in $W$ (which is not necessarily a ball --- parts of $\bd X_a$ may have been glued together). |
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|
978 |
Define a {\it permissible decomposition} of $W$ to be a map |
108 | 979 |
\[ |
475
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|
980 |
\coprod_a X_a \to W, |
108 | 981 |
\] |
475
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changeset
|
982 |
which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$. |
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changeset
|
983 |
Roughly, a permissible decomposition is like a ball decomposition where we don't care in which order the balls |
07c18e2abd8f
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changeset
|
984 |
are glued up to yield $W$, so long as there is some (non-pathological) way to glue them. |
07c18e2abd8f
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diff
changeset
|
985 |
|
479
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476
diff
changeset
|
986 |
Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement |
475
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changeset
|
987 |
of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$ |
07c18e2abd8f
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changeset
|
988 |
with $\du_b Y_b = M_i$ for some $i$. |
191
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|
989 |
|
419
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changeset
|
990 |
\begin{defn} |
479
cfad13b6b1e5
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diff
changeset
|
991 |
The poset $\cell(W)$ has objects the permissible decompositions of $W$, |
340
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|
992 |
and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
191
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|
993 |
See Figure \ref{partofJfig} for an example. |
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|
994 |
\end{defn} |
119 | 995 |
|
996 |
\begin{figure}[!ht] |
|
997 |
\begin{equation*} |
|
222
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diff
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|
998 |
\mathfig{.63}{ncat/zz2} |
119 | 999 |
\end{equation*} |
329
eb03c4a92f98
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|
1000 |
\caption{A small part of $\cell(W)$} |
119 | 1001 |
\label{partofJfig} |
1002 |
\end{figure} |
|
1003 |
||
191
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|
1004 |
An $n$-category $\cC$ determines |
329
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|
1005 |
a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
108 | 1006 |
(possibly with additional structure if $k=n$). |
197 | 1007 |
Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
1008 |
and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries |
|
1009 |
are splittable along this decomposition. |
|
108 | 1010 |
|
191
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|
1011 |
\begin{defn} |
329
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|
1012 |
Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
494
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|
1013 |
For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset |
191
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|
1014 |
\begin{equation} |
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|
1015 |
\label{eq:psi-C} |
197 | 1016 |
\psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl |
191
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|
1017 |
\end{equation} |
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|
1018 |
where the restrictions to the various pieces of shared boundaries amongst the cells |
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|
1019 |
$X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells). |
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|
1020 |
If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$. |
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|
1021 |
\end{defn} |
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|
1022 |
|
419
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|
1023 |
If $k=n$ in the above definition and we are enriching in some auxiliary category, |
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changeset
|
1024 |
we need to say a bit more. |
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changeset
|
1025 |
We can rewrite Equation \ref{eq:psi-C} as |
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|
1026 |
\begin{equation} \label{eq:psi-CC} |
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changeset
|
1027 |
\psi_{\cC;W}(x) \deq \coprod_\beta \prod_a \cC(X_a; \beta) , |
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changeset
|
1028 |
\end{equation} |
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changeset
|
1029 |
where $\beta$ runs through labelings of the $k{-}1$-skeleton of the decomposition |
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changeset
|
1030 |
(which are compatible when restricted to the $k{-}2$-skeleton), and $\cC(X_a; \beta)$ |
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changeset
|
1031 |
means the subset of $\cC(X_a)$ whose restriction to $\bd X_a$ agress with $\beta$. |
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changeset
|
1032 |
If we are enriching over $\cS$ and $k=n$, then $\cC(X_a; \beta)$ is an object in |
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changeset
|
1033 |
$\cS$ and the coproduct and product in Equation \ref{eq:psi-CC} should be replaced by the approriate |
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|
1034 |
operations in $\cS$ (e.g. direct sum and tensor product if $\cS$ is Vect). |
191
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1035 |
|
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|
1036 |
Finally, we construct $\cl{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$: |
191
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1037 |
|
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1038 |
\begin{defn}[System of fields functor] |
415
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|
1039 |
\label{def:colim-fields} |
402 | 1040 |
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
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|
1041 |
That is, for each decomposition $x$ there is a map |
402 | 1042 |
$\psi_{\cC;W}(x)\to \cl{\cC}(W)$, these maps are compatible with the refinement maps |
1043 |
above, and $\cl{\cC}(W)$ is universal with respect to these properties. |
|
191
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|
1044 |
\end{defn} |
112 | 1045 |
|
191
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|
1046 |
\begin{defn}[System of fields functor, $A_\infty$ case] |
402 | 1047 |
When $\cC$ is an $A_\infty$ $n$-category, $\cl{\cC}(W)$ for $W$ a $k$-manifold with $k < n$ |
340
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1048 |
is defined as above, as the colimit of $\psi_{\cC;W}$. |
402 | 1049 |
When $W$ is an $n$-manifold, the chain complex $\cl{\cC}(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$. |
191
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|
1050 |
\end{defn} |
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|
1051 |
|
402 | 1052 |
We can specify boundary data $c \in \cl{\cC}(\bdy W)$, and define functors $\psi_{\cC;W,c}$ |
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|
1053 |
with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$. |
111 | 1054 |
|
422 | 1055 |
We now give more concrete descriptions of the above colimits. |
1056 |
||
1057 |
In the non-enriched case (e.g.\ $k<n$), where each $\cC(X_a; \beta)$ is just a set, |
|
1058 |
the colimit is |
|
1059 |
\[ |
|
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|
1060 |
\cl{\cC}(W,c) = \left( \coprod_x \coprod_\beta \prod_a \cC(X_a; \beta) \right) \Bigg/ \sim , |
422 | 1061 |
\] |
1062 |
where $x$ runs through decomposition of $W$, and $\sim$ is the obvious equivalence relation |
|
1063 |
induced by refinement and gluing. |
|
1064 |
If $\cC$ is enriched over vector spaces and $W$ is an $n$-manifold, |
|
1065 |
we can take |
|
191
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|
1066 |
\begin{equation*} |
494
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|
1067 |
\cl{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) \Bigg/ K, |
191
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|
1068 |
\end{equation*} |
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|
1069 |
where $K$ is the vector space spanned by elements $a - g(a)$, with |
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|
1070 |
$a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
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|
1071 |
\to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
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|
1072 |
|
225
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|
1073 |
In the $A_\infty$ case, enriched over chain complexes, the concrete description of the homotopy colimit |
197 | 1074 |
is more involved. |
542
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|
1075 |
We will describe two different (but homotopy equivalent) versions of the homotopy colimit of $\psi_{\cC;W}$. |
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|
1076 |
The first is the usual one, which works for any indexing category. |
550 | 1077 |
The second construction, which we call the {\it local} homotopy colimit, |
542
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|
1078 |
is more closely related to the blob complex |
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|
1079 |
construction of \S \ref{sec:blob-definition} and takes advantage of local (gluing) properties |
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|
1080 |
of the indexing category $\cell(W)$. |
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|
1081 |
|
191
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|
1082 |
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
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|
1083 |
Such sequences (for all $m$) form a simplicial set in $\cell(W)$. |
402 | 1084 |
Define $\cl{\cC}(W)$ as a vector space via |
112 | 1085 |
\[ |
402 | 1086 |
\cl{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
112 | 1087 |
\] |
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|
1088 |
where the sum is over all $m$ and all $m$-sequences $(x_i)$, and each summand is degree shifted by $m$. |
463 | 1089 |
Elements of a summand indexed by an $m$-sequence will be call $m$-simplices. |
402 | 1090 |
We endow $\cl{\cC}(W)$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ |
112 | 1091 |
summands plus another term using the differential of the simplicial set of $m$-sequences. |
1092 |
More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
|
402 | 1093 |
summand of $\cl{\cC}(W)$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
112 | 1094 |
\[ |
191
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|
1095 |
\bd (a, \bar{x}) = (\bd a, \bar{x}) + (-1)^{\deg{a}} (g(a), d_0(\bar{x})) + (-1)^{\deg{a}} \sum_{j=1}^k (-1)^{j} (a, d_j(\bar{x})) , |
112 | 1096 |
\] |
1097 |
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 | 1098 |
is the usual gluing map coming from the antirefinement $x_0 \le x_1$. |
422 | 1099 |
%\nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which |
1100 |
%combine only two balls at a time; for $n=1$ this version will lead to usual definition |
|
1101 |
%of $A_\infty$ category} |
|
108 | 1102 |
|
113 | 1103 |
We can think of this construction as starting with a disjoint copy of a complex for each |
461
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|
1104 |
permissible decomposition (the 0-simplices). |
113 | 1105 |
Then we glue these together with mapping cylinders coming from gluing maps |
461
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|
1106 |
(the 1-simplices). |
340
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|
1107 |
Then we kill the extra homology we just introduced with mapping |
461
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|
1108 |
cylinders between the mapping cylinders (the 2-simplices), and so on. |
113 | 1109 |
|
542
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|
1110 |
Next we describe the local homotopy colimit. |
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|
1111 |
This is similar to the usual homotopy colimit, but using |
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|
1112 |
a cone-product set (Remark \ref{blobsset-remark}) in place of a simplicial set. |
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|
1113 |
The cone-product $m$-polyhedra for the set are pairs $(x, E)$, where $x$ is a decomposition of $W$ |
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|
1114 |
and $E$ is an $m$-blob diagram such that each blob is a union of balls of $x$. |
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|
1115 |
(Recall that this means that the interiors of |
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|
1116 |
each pair of blobs (i.e.\ balls) of $E$ are either disjoint or nested.) |
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|
1117 |
To each $(x, E)$ we associate the chain complex $\psi_{\cC;W}(x)$, shifted in degree by $m$. |
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|
1118 |
The boundary has a term for omitting each blob of $E$. |
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|
1119 |
If we omit an innermost blob then we replace $x$ by the formal difference $x - \gl(x)$, where |
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|
1120 |
$\gl(x)$ is obtained from $x$ by gluing together the balls of $x$ contained in the blob we are omitting. |
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|
1121 |
The gluing maps of $\cC$ give us a maps from $\psi_{\cC;W}(x)$ to $\psi_{\cC;W}(\gl(x))$. |
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|
1122 |
|
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|
1123 |
One can show that the usual hocolimit and the local hocolimit are homotopy equivalent using an |
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|
1124 |
Eilenberg-Zilber type subdivision argument. |
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|
1125 |
|
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|
1126 |
\medskip |
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|
1127 |
|
552 | 1128 |
$\cl{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
1129 |
Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}. |
|
108 | 1130 |
|
420 | 1131 |
It is easy to see that |
422 | 1132 |
there are well-defined maps $\cl{\cC}(W)\to\cl{\cC}(\bd W)$, and that these maps |
108 | 1133 |
comprise a natural transformation of functors. |
1134 |
||
415
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|
1135 |
\begin{lem} |
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|
1136 |
\label{lem:colim-injective} |
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|
1137 |
Let $W$ be a manifold of dimension less than $n$. Then for each |
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|
1138 |
decomposition $x$ of $W$ the natural map $\psi_{\cC;W}(x)\to \cl{\cC}(W)$ is injective. |
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|
1139 |
\end{lem} |
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|
1140 |
\begin{proof} |
531
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|
1141 |
$\cl{\cC}(W)$ is a colimit of a diagram of sets, and each of the arrows in the diagram is |
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changeset
|
1142 |
injective. |
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changeset
|
1143 |
Concretely, the colimit is the disjoint union of the sets (one for each decomposition of $W$), |
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|
1144 |
modulo the relation which identifies the domain of each of the injective maps |
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changeset
|
1145 |
with it's image. |
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changeset
|
1146 |
|
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diff
changeset
|
1147 |
To save ink and electrons we will simplify notation and write $\psi(x)$ for $\psi_{\cC;W}(x)$. |
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|
1148 |
|
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|
1149 |
Suppose $a, \hat{a}\in \psi(x)$ have the same image in $\cl{\cC}(W)$ but $a\ne \hat{a}$. |
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|
1150 |
Then there exist |
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|
1151 |
\begin{itemize} |
da9bf150bf3d
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changeset
|
1152 |
\item decompositions $x = x_0, x_1, \ldots , x_{k-1}, x_k = x$ and $v_1,\ldots, v_k$ of $W$; |
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|
1153 |
\item anti-refinements $v_i\to x_i$ and $v_i\to x_{i-1}$; and |
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changeset
|
1154 |
\item elements $a_i\in \psi(x_i)$ and $b_i\in \psi(v_i)$, with $a_0 = a$ and $a_k = \hat{a}$, |
da9bf150bf3d
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diff
changeset
|
1155 |
such that $b_i$ and $b_{i+1}$both map to (glue up to) $a_i$. |
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changeset
|
1156 |
\end{itemize} |
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changeset
|
1157 |
In other words, we have a zig-zag of equivalences starting at $a$ and ending at $\hat{a}$. |
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changeset
|
1158 |
The idea of the proof is to produce a similar zig-zag where everything antirefines to the same |
535
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|
1159 |
disjoint union of balls, and then invoke Axiom \ref{nca-assoc} which ensures associativity. |
531
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diff
changeset
|
1160 |
|
da9bf150bf3d
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changeset
|
1161 |
Let $z$ be a decomposition of $W$ which is in general position with respect to all of the |
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changeset
|
1162 |
$x_i$'s and $v_i$'s. |
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changeset
|
1163 |
There there decompositions $x'_i$ and $v'_i$ (for all $i$) such that |
da9bf150bf3d
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diff
changeset
|
1164 |
\begin{itemize} |
da9bf150bf3d
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|
1165 |
\item $x'_i$ antirefines to $x_i$ and $z$; |
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diff
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|
1166 |
\item $v'_i$ antirefines to $x'_i$, $x'_{i-1}$ and $v_i$; |
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|
1167 |
\item $b_i$ is the image of some $b'_i\in \psi(v'_i)$; and |
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diff
changeset
|
1168 |
\item $a_i$ is the image of some $a'_i\in \psi(x'_i)$, which in turn is the image |
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diff
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|
1169 |
of $b'_i$ and $b'_{i+1}$. |
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|
1170 |
\end{itemize} |
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changeset
|
1171 |
Now consider the diagrams |
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changeset
|
1172 |
\[ \xymatrix{ |
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|
1173 |
& \psi(x'_{i-1}) \ar[rd] & \\ |
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diff
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|
1174 |
\psi(v'_i) \ar[ru] \ar[rd] & & \psi(z) \\ |
da9bf150bf3d
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|
1175 |
& \psi(x'_i) \ar[ru] & |
da9bf150bf3d
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diff
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|
1176 |
} \] |
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diff
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|
1177 |
The associativity axiom applied to this diagram implies that $a'_{i-1}$ and $a'_i$ |
da9bf150bf3d
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diff
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|
1178 |
map to the same element $c\in \psi(z)$. |
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diff
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|
1179 |
Therefore $a'_0$ and $a'_k$ both map to $c$. |
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diff
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|
1180 |
But $a'_0$ and $a'_k$ are both elements of $\psi(x'_0)$ (because $x'_k = x'_0$). |
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diff
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|
1181 |
So by the injectivity clause of the composition axiom, we must have that $a'_0 = a'_k$. |
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|
1182 |
But this implies that $a = a_0 = a_k = \hat{a}$, contrary to our assumption that $a\ne \hat{a}$. |
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|
1183 |
\end{proof} |
402 | 1184 |
|
552 | 1185 |
%\nn{need to finish explaining why we have a system of fields; |
1186 |
%define $k$-cat $\cC(\cdot\times W)$} |
|
108 | 1187 |
|
1188 |
\subsection{Modules} |
|
95 | 1189 |
|
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|
1190 |
Next we define ordinary and $A_\infty$ $n$-category modules. |
199 | 1191 |
The definition will be very similar to that of $n$-categories, |
1192 |
but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
|
198 | 1193 |
|
104 | 1194 |
Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
102 | 1195 |
in the context of an $m{+}1$-dimensional TQFT. |
1196 |
Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
|
1197 |
This will be explained in more detail as we present the axioms. |
|
1198 |
||
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|
1199 |
Throughout, we fix an $n$-category $\cC$. |
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|
1200 |
For all but one axiom, it doesn't matter whether $\cC$ is an ordinary $n$-category or an $A_\infty$ $n$-category. |
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|
1201 |
We state the final axiom, regarding actions of homeomorphisms, differently in the two cases. |
102 | 1202 |
|
1203 |
Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair |
|
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1204 |
$$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$ |
102 | 1205 |
We call $B$ the ball and $N$ the marking. |
1206 |
A homeomorphism between marked $k$-balls is a homeomorphism of balls which |
|
1207 |
restricts to a homeomorphism of markings. |
|
1208 |
||
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|
1209 |
\begin{module-axiom}[Module morphisms] \label{module-axiom-funct} |
102 | 1210 |
{For each $0 \le k \le n$, we have a functor $\cM_k$ from |
1211 |
the category of marked $k$-balls and |
|
1212 |
homeomorphisms to the category of sets and bijections.} |
|
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|
1213 |
\end{module-axiom} |
102 | 1214 |
|
1215 |
(As with $n$-categories, we will usually omit the subscript $k$.) |
|
1216 |
||
423 | 1217 |
For example, let $\cD$ be the TQFT which assigns to a $k$-manifold $N$ the set |
1218 |
of maps from $N$ to $T$ (for $k\le m$), modulo homotopy (and possibly linearized) if $k=m$. |
|
104 | 1219 |
Let $W$ be an $(m{-}n{+}1)$-dimensional manifold with boundary. |
1220 |
Let $\cC$ be the $n$-category with $\cC(X) \deq \cD(X\times \bd W)$. |
|
423 | 1221 |
Let $\cM(B, N) \deq \cD((B\times \bd W)\cup (N\times W))$ |
1222 |
(see Example \ref{ex:maps-with-fiber}). |
|
104 | 1223 |
(The union is along $N\times \bd W$.) |
423 | 1224 |
%(If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be |
1225 |
%the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.) |
|
102 | 1226 |
|
182 | 1227 |
\begin{figure}[!ht] |
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|
1228 |
$$\mathfig{.55}{ncat/boundary-collar}$$ |
182 | 1229 |
\caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure} |
1230 |
||
103 | 1231 |
Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
1232 |
Call such a thing a {marked $k{-}1$-hemisphere}. |
|
102 | 1233 |
|
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|
1234 |
\begin{lem} |
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|
1235 |
\label{lem:hemispheres} |
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|
1236 |
{For each $0 \le k \le n-1$, we have a functor $\cl\cM_k$ from |
104 | 1237 |
the category of marked $k$-hemispheres and |
102 | 1238 |
homeomorphisms to the category of sets and bijections.} |
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|
1239 |
\end{lem} |
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|
1240 |
The proof is exactly analogous to that of Lemma \ref{lem:spheres}, and we omit the details. |
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|
1241 |
We use the same type of colimit construction. |
102 | 1242 |
|
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|
1243 |
In our example, $\cl\cM(H) = \cD(H\times\bd W \cup \bd H\times W)$. |
104 | 1244 |
|
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|
1245 |
\begin{module-axiom}[Module boundaries (maps)] |
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|
1246 |
{For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cl\cM(\bd M)$. |
102 | 1247 |
These maps, for various $M$, comprise a natural transformation of functors.} |
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|
1248 |
\end{module-axiom} |
102 | 1249 |
|
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|
1250 |
Given $c\in\cl\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
102 | 1251 |
|
1252 |
If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
|
1253 |
then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$ |
|
1254 |
and $c\in \cC(\bd M)$. |
|
1255 |
||
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|
1256 |
\begin{lem}[Boundary from domain and range] |
423 | 1257 |
{Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k{-}1$-hemisphere ($1\le k\le n$), |
1258 |
$M_i$ is a marked $k{-}1$-ball, and $E = M_1\cap M_2$ is a marked $k{-}2$-hemisphere. |
|
104 | 1259 |
Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the |
424
6ebf92d2ccef
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423
diff
changeset
|
1260 |
two maps $\bd: \cM(M_i)\to \cl\cM(E)$. |
423 | 1261 |
Then we have an injective map |
102 | 1262 |
\[ |
424
6ebf92d2ccef
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diff
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|
1263 |
\gl_E : \cM(M_1) \times_{\cl\cM(E)} \cM(M_2) \hookrightarrow \cl\cM(H) |
102 | 1264 |
\] |
1265 |
which is natural with respect to the actions of homeomorphisms.} |
|
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|
1266 |
\end{lem} |
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|
1267 |
Again, this is in exact analogy with Lemma \ref{lem:domain-and-range}. |
102 | 1268 |
|
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|
1269 |
Let $\cl\cM(H)_E$ denote the image of $\gl_E$. |
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diff
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|
1270 |
We will refer to elements of $\cl\cM(H)_E$ as ``splittable along $E$" or ``transverse to $E$". |
110 | 1271 |
|
424
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diff
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|
1272 |
\begin{lem}[Module to category restrictions] |
103 | 1273 |
{For each marked $k$-hemisphere $H$ there is a restriction map |
424
6ebf92d2ccef
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diff
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|
1274 |
$\cl\cM(H)\to \cC(H)$. |
103 | 1275 |
($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) |
1276 |
These maps comprise a natural transformation of functors.} |
|
424
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|
1277 |
\end{lem} |
102 | 1278 |
|
103 | 1279 |
Note that combining the various boundary and restriction maps above |
110 | 1280 |
(for both modules and $n$-categories) |
103 | 1281 |
we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$ |
1282 |
a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$. |
|
110 | 1283 |
The subset is the subset of morphisms which are appropriately splittable (transverse to the |
1284 |
cutting submanifolds). |
|
103 | 1285 |
This fact will be used below. |
102 | 1286 |
|
104 | 1287 |
In our example, the various restriction and gluing maps above come from |
1288 |
restricting and gluing maps into $T$. |
|
1289 |
||
1290 |
We require two sorts of composition (gluing) for modules, corresponding to two ways |
|
103 | 1291 |
of splitting a marked $k$-ball into two (marked or plain) $k$-balls. |
119 | 1292 |
(See Figure \ref{zzz3}.) |
103 | 1293 |
|
119 | 1294 |
\begin{figure}[!ht] |
1295 |
\begin{equation*} |
|
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|
1296 |
\mathfig{.4}{ncat/zz3} |
119 | 1297 |
\end{equation*} |
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|
1298 |
\caption{Module composition (top); $n$-category action (bottom).} |
119 | 1299 |
\label{zzz3} |
1300 |
\end{figure} |
|
1301 |
||
1302 |
First, we can compose two module morphisms to get another module morphism. |
|
103 | 1303 |
|
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|
1304 |
\begin{module-axiom}[Module composition] |
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|
1305 |
{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 | 1306 |
and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball. |
1307 |
Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere. |
|
1308 |
Note that each of $M$, $M_1$ and $M_2$ has its boundary split into two marked $k{-}1$-balls by $E$. |
|
1309 |
We have restriction (domain or range) maps $\cM(M_i)_E \to \cM(Y)$. |
|
1310 |
Let $\cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E$ denote the fibered product of these two maps. |
|
1311 |
Then (axiom) we have a map |
|
1312 |
\[ |
|
1313 |
\gl_Y : \cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E \to \cM(M)_E |
|
1314 |
\] |
|
1315 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
1316 |
to the intersection of the boundaries of $M$ and $M_i$. |
|
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|
1317 |
If $k < n$, |
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changeset
|
1318 |
or if $k=n$ and we are in the $A_\infty$ case, |
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|
1319 |
we require that $\gl_Y$ is injective. |
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|
1320 |
(For $k=n$ in the ordinary (non-$A_\infty$) case, see below.)} |
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|
1321 |
\end{module-axiom} |
119 | 1322 |
|
1323 |
||
103 | 1324 |
Second, we can compose an $n$-category morphism with a module morphism to get another |
1325 |
module morphism. |
|
1326 |
We'll call this the action map to distinguish it from the other kind of composition. |
|
1327 |
||
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|
1328 |
\begin{module-axiom}[$n$-category action] |
103 | 1329 |
{Let $M = X \cup_Y M'$, where $M$ and $M'$ are marked $k$-balls ($0\le k\le n$), |
1330 |
$X$ is a plain $k$-ball, |
|
1331 |
and $Y = X\cap M'$ is a $k{-}1$-ball. |
|
1332 |
Let $E = \bd Y$, which is a $k{-}2$-sphere. |
|
1333 |
We have restriction maps $\cM(M')_E \to \cC(Y)$ and $\cC(X)_E\to \cC(Y)$. |
|
1334 |
Let $\cC(X)_E \times_{\cC(Y)} \cM(M')_E$ denote the fibered product of these two maps. |
|
1335 |
Then (axiom) we have a map |
|
1336 |
\[ |
|
1337 |
\gl_Y :\cC(X)_E \times_{\cC(Y)} \cM(M')_E \to \cM(M)_E |
|
1338 |
\] |
|
1339 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
1340 |
to the intersection of the boundaries of $X$ and $M'$. |
|
546
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|
1341 |
If $k < n$, |
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parents:
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diff
changeset
|
1342 |
or if $k=n$ and we are in the $A_\infty$ case, |
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|
1343 |
we require that $\gl_Y$ is injective. |
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changeset
|
1344 |
(For $k=n$ in the ordinary (non-$A_\infty$) case, see below.)} |
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|
1345 |
\end{module-axiom} |
103 | 1346 |
|
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|
1347 |
\begin{module-axiom}[Strict associativity] |
423 | 1348 |
The composition and action maps above are strictly associative. |
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|
1349 |
Given any decomposition of a large marked ball into smaller marked and unmarked balls |
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|
1350 |
any sequence of pairwise gluings yields (via composition and action maps) the same result. |
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|
1351 |
\end{module-axiom} |
103 | 1352 |
|
110 | 1353 |
Note that the above associativity axiom applies to mixtures of module composition, |
1354 |
action maps and $n$-category composition. |
|
119 | 1355 |
See Figure \ref{zzz1b}. |
1356 |
||
1357 |
\begin{figure}[!ht] |
|
1358 |
\begin{equation*} |
|
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|
1359 |
\mathfig{0.49}{ncat/zz0} \mathfig{0.49}{ncat/zz1} |
119 | 1360 |
\end{equation*} |
1361 |
\caption{Two examples of mixed associativity} |
|
1362 |
\label{zzz1b} |
|
1363 |
\end{figure} |
|
1364 |
||
110 | 1365 |
|
1366 |
The above three axioms are equivalent to the following axiom, |
|
103 | 1367 |
which we state in slightly vague form. |
1368 |
||
1369 |
\xxpar{Module multi-composition:} |
|
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|
1370 |
{Given any splitting |
103 | 1371 |
\[ |
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|
1372 |
X_1 \sqcup\cdots\sqcup X_p \sqcup M_1\sqcup\cdots\sqcup M_q \to M |
103 | 1373 |
\] |
1374 |
of a marked $k$-ball $M$ |
|
1375 |
into small (marked and plain) $k$-balls $M_i$ and $X_j$, there is a |
|
1376 |
map from an appropriate subset (like a fibered product) |
|
1377 |
of |
|
1378 |
\[ |
|
1379 |
\cC(X_1)\times\cdots\times\cC(X_p) \times \cM(M_1)\times\cdots\times\cM(M_q) |
|
1380 |
\] |
|
1381 |
to $\cM(M)$, |
|
1382 |
and these various multifold composition maps satisfy an |
|
1383 |
operad-type strict associativity condition.} |
|
1384 |
||
423 | 1385 |
The above operad-like structure is analogous to the swiss cheese operad |
1386 |
\cite{MR1718089}. |
|
1387 |
||
1388 |
\medskip |
|
1389 |
||
1390 |
We can define marked pinched products $\pi:E\to M$ of marked balls analogously to the |
|
1391 |
plain ball case. |
|
1392 |
Note that a marked pinched product can be decomposed into either |
|
1393 |
two marked pinched products or a plain pinched product and a marked pinched product. |
|
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|
1394 |
%\nn{should maybe give figure} |
103 | 1395 |
|
423 | 1396 |
\begin{module-axiom}[Product (identity) morphisms] |
1397 |
For each pinched product $\pi:E\to M$, with $M$ a marked $k$-ball and $E$ a marked |
|
1398 |
$k{+}m$-ball ($m\ge 1$), |
|
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|
1399 |
there is a map $\pi^*:\cM(M)\to \cM(E)$. |
423 | 1400 |
These maps must satisfy the following conditions. |
1401 |
\begin{enumerate} |
|
1402 |
\item |
|
1403 |
If $\pi:E\to M$ and $\pi':E'\to M'$ are marked pinched products, and |
|
1404 |
if $f:M\to M'$ and $\tilde{f}:E \to E'$ are maps such that the diagram |
|
103 | 1405 |
\[ \xymatrix{ |
423 | 1406 |
E \ar[r]^{\tilde{f}} \ar[d]_{\pi} & E' \ar[d]^{\pi'} \\ |
103 | 1407 |
M \ar[r]^{f} & M' |
1408 |
} \] |
|
423 | 1409 |
commutes, then we have |
1410 |
\[ |
|
1411 |
\pi'^*\circ f = \tilde{f}\circ \pi^*. |
|
1412 |
\] |
|
1413 |
\item |
|
1414 |
Product morphisms are compatible with module composition and module action. |
|
1415 |
Let $\pi:E\to M$, $\pi_1:E_1\to M_1$, and $\pi_2:E_2\to M_2$ |
|
1416 |
be pinched products with $E = E_1\cup E_2$. |
|
1417 |
Let $a\in \cM(M)$, and let $a_i$ denote the restriction of $a$ to $M_i\sub M$. |
|
1418 |
Then |
|
1419 |
\[ |
|
1420 |
\pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) . |
|
1421 |
\] |
|
1422 |
Similarly, if $\rho:D\to X$ is a pinched product of plain balls and |
|
1423 |
$E = D\cup E_1$, then |
|
1424 |
\[ |
|
1425 |
\pi^*(a) = \rho^*(a')\bullet \pi_1^*(a_1), |
|
1426 |
\] |
|
1427 |
where $a'$ is the restriction of $a$ to $D$. |
|
1428 |
\item |
|
1429 |
Product morphisms are associative. |
|
1430 |
If $\pi:E\to M$ and $\rho:D\to E$ are marked pinched products then |
|
1431 |
\[ |
|
1432 |
\rho^*\circ\pi^* = (\pi\circ\rho)^* . |
|
1433 |
\] |
|
1434 |
\item |
|
1435 |
Product morphisms are compatible with restriction. |
|
1436 |
If we have a commutative diagram |
|
1437 |
\[ \xymatrix{ |
|
1438 |
D \ar@{^(->}[r] \ar[d]_{\rho} & E \ar[d]^{\pi} \\ |
|
1439 |
Y \ar@{^(->}[r] & M |
|
1440 |
} \] |
|
1441 |
such that $\rho$ and $\pi$ are pinched products, then |
|
1442 |
\[ |
|
1443 |
\res_D\circ\pi^* = \rho^*\circ\res_Y . |
|
1444 |
\] |
|
1445 |
($Y$ could be either a marked or plain ball.) |
|
1446 |
\end{enumerate} |
|
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|
1447 |
\end{module-axiom} |
103 | 1448 |
|
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|
1449 |
As in the $n$-category definition, once we have product morphisms we can define |
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|
1450 |
collar maps $\cM(M)\to \cM(M)$. |
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|
1451 |
Note that there are two cases: |
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|
1452 |
the collar could intersect the marking of the marked ball $M$, in which case |
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|
1453 |
we use a product on a morphism of $\cM$; or the collar could be disjoint from the marking, |
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|
1454 |
in which case we use a product on a morphism of $\cC$. |
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|
1455 |
|
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|
1456 |
In our example, elements $a$ of $\cM(M)$ maps to $T$, and $\pi^*(a)$ is the pullback of |
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|
1457 |
$a$ along a map associated to $\pi$. |
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|
1458 |
|
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|
1459 |
\medskip |
110 | 1460 |
|
103 | 1461 |
There are two alternatives for the next axiom, according whether we are defining |
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|
1462 |
modules for ordinary $n$-categories or $A_\infty$ $n$-categories. |
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|
1463 |
In the ordinary case we require |
103 | 1464 |
|
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|
1465 |
\begin{module-axiom}[\textup{\textbf{[ordinary version]}} Extended isotopy invariance in dimension $n$] |
103 | 1466 |
{Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts |
424
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|
1467 |
to the identity on $\bd M$ and is isotopic (rel boundary) to the identity. |
103 | 1468 |
Then $f$ acts trivially on $\cM(M)$.} |
424
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|
1469 |
In addition, collar maps act trivially on $\cM(M)$. |
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|
1470 |
\end{module-axiom} |
103 | 1471 |
|
1472 |
We emphasize that the $\bd M$ above means boundary in the marked $k$-ball sense. |
|
1473 |
In other words, if $M = (B, N)$ then we require only that isotopies are fixed |
|
1474 |
on $\bd B \setmin N$. |
|
1475 |
||
1476 |
For $A_\infty$ modules we require |
|
1477 |
||
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|
1478 |
%\addtocounter{module-axiom}{-1} |
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|
1479 |
\begin{module-axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act] |
424
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|
1480 |
For each marked $n$-ball $M$ and each $c\in \cM(\bd M)$ we have a map of chain complexes |
103 | 1481 |
\[ |
1482 |
C_*(\Homeo_\bd(M))\ot \cM(M; c) \to \cM(M; c) . |
|
1483 |
\] |
|
1484 |
Here $C_*$ means singular chains and $\Homeo_\bd(M)$ is the space of homeomorphisms of $M$ |
|
1485 |
which fix $\bd M$. |
|
437 | 1486 |
These action maps are required to be associative up to homotopy, as in Theorem \ref{thm:CH-associativity}, |
424
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|
1487 |
and also compatible with composition (gluing) in the sense that |
437 | 1488 |
a diagram like the one in Theorem \ref{thm:CH} commutes. |
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|
1489 |
\end{module-axiom} |
103 | 1490 |
|
424
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|
1491 |
As with the $n$-category version of the above axiom, we should also have families of collar maps act. |
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|
1492 |
|
103 | 1493 |
\medskip |
102 | 1494 |
|
104 | 1495 |
Note that the above axioms imply that an $n$-category module has the structure |
1496 |
of an $n{-}1$-category. |
|
1497 |
More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$, |
|
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|
1498 |
where $X$ is a $k$-ball and in the product $X\times J$ we pinch |
104 | 1499 |
above the non-marked boundary component of $J$. |
200 | 1500 |
(More specifically, we collapse $X\times P$ to a single point, where |
1501 |
$P$ is the non-marked boundary component of $J$.) |
|
104 | 1502 |
Then $\cE$ has the structure of an $n{-}1$-category. |
102 | 1503 |
|
105 | 1504 |
All marked $k$-balls are homeomorphic, unless $k = 1$ and our manifolds |
1505 |
are oriented or Spin (but not unoriented or $\text{Pin}_\pm$). |
|
1506 |
In this case ($k=1$ and oriented or Spin), there are two types |
|
1507 |
of marked 1-balls, call them left-marked and right-marked, |
|
1508 |
and hence there are two types of modules, call them right modules and left modules. |
|
1509 |
In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$), |
|
1510 |
there is no left/right module distinction. |
|
1511 |
||
130 | 1512 |
\medskip |
1513 |
||
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|
1514 |
We now give some examples of modules over ordinary and $A_\infty$ $n$-categories. |
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|
1515 |
|
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|
1516 |
\begin{example}[Examples from TQFTs] |
425 | 1517 |
\rm |
1518 |
Continuing Example \ref{ex:ncats-from-tqfts}, with $\cF$ a TQFT, $W$ an $n{-}j$-manifold, |
|
1519 |
and $\cF(W)$ the $j$-category associated to $W$. |
|
1520 |
Let $Y$ be an $(n{-}j{+}1)$-manifold with $\bd Y = W$. |
|
1521 |
Define a $\cF(W)$ module $\cF(Y)$ as follows. |
|
1522 |
If $M = (B, N)$ is a marked $k$-ball with $k<j$ let |
|
1523 |
$\cF(Y)(M)\deq \cF((B\times W) \cup (N\times Y))$. |
|
1524 |
If $M = (B, N)$ is a marked $j$-ball and $c\in \cl{\cF(Y)}(\bd M)$ let |
|
1525 |
$\cF(Y)(M)\deq A_\cF((B\times W) \cup (N\times Y); c)$. |
|
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|
1526 |
\end{example} |
108 | 1527 |
|
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|
1528 |
\begin{example}[Examples from the blob complex] \label{bc-module-example} |
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|
1529 |
\rm |
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|
1530 |
In the previous example, we can instead define |
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|
1531 |
$\cF(Y)(M)\deq \bc_*((B\times W) \cup (N\times Y), c; \cF)$ (when $\dim(M) = n$) |
448
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|
1532 |
and get a module for the $A_\infty$ $n$-category associated to $\cF$ as in |
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|
1533 |
Example \ref{ex:blob-complexes-of-balls}. |
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|
1534 |
\end{example} |
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|
1535 |
|
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|
1536 |
|
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|
1537 |
\begin{example} |
425 | 1538 |
\rm |
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|
1539 |
Suppose $S$ is a topological space, with a subspace $T$. |
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|
1540 |
We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$ |
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|
1541 |
for $k<n$ the set $\pi_{\leq n}(S,T)(B,N)$ consists of all continuous maps of pairs |
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|
1542 |
$(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all |
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|
1543 |
such maps modulo homotopies fixed on $\bdy B \setminus N$. |
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|
1544 |
This is a module over the fundamental $n$-category $\pi_{\leq n}(S)$ of $S$, from Example \ref{ex:maps-to-a-space}. |
420 | 1545 |
\end{example} |
340
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|
1546 |
Modifications corresponding to Examples \ref{ex:maps-to-a-space-with-a-fiber} and |
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|
1547 |
\ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to |
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|
1548 |
Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains. |
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|
1549 |
|
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|
1550 |
|
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|
1551 |
|
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|
1552 |
|
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|
1553 |
|
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|
1554 |
\subsection{Modules as boundary labels (colimits for decorated manifolds)} |
112 | 1555 |
\label{moddecss} |
108 | 1556 |
|
685
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|
1557 |
Fix an ordinary $n$-category or $A_\infty$ $n$-category $\cC$. |
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|
1558 |
Let $W$ be a $k$-manifold ($k\le n$), |
143 | 1559 |
let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$, |
1560 |
and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$. |
|
1561 |
||
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|
1562 |
We will define a set $\cC(W, \cN)$ using a colimit construction very similar to |
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the one appearing in \S \ref{ss:ncat_fields} above. |
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|
1564 |
(If $k = n$ and our $n$-categories are enriched, then |
108 | 1565 |
$\cC(W, \cN)$ will have additional structure; see below.) |
1566 |
||
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|
1567 |
Define a permissible decomposition of $W$ to be a map |
108 | 1568 |
\[ |
494
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|
1569 |
\left(\bigsqcup_a X_a\right) \sqcup \left(\bigsqcup_{i,b} M_{ib}\right) \to W, |
108 | 1570 |
\] |
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|
1571 |
where each $X_a$ is a plain $k$-ball disjoint, in $W$, from $\cup Y_i$, and |
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|
1572 |
each $M_{ib}$ is a marked $k$-ball intersecting $Y_i$ (once mapped into $W$), |
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|
1573 |
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 | 1574 |
(See Figure \ref{mblabel}.) |
435 | 1575 |
\begin{figure}[t] |
1576 |
\begin{equation*} |
|
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|
1577 |
\mathfig{.4}{ncat/mblabel} |
435 | 1578 |
\end{equation*} |
1579 |
\caption{A permissible decomposition of a manifold |
|
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1580 |
whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$. |
435 | 1581 |
Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel} |
1582 |
\end{figure} |
|
108 | 1583 |
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
1584 |
of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
|
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|
1585 |
This defines a partial ordering $\cell(W)$, which we will think of as a category. |
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|
1586 |
(The objects of $\cell(D)$ are permissible decompositions of $W$, and there is a unique |
108 | 1587 |
morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
1588 |
||
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|
1589 |
The collection of modules $\cN$ determines |
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|
1590 |
a functor $\psi_\cN$ from $\cell(W)$ to the category of sets |
108 | 1591 |
(possibly with additional structure if $k=n$). |
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|
1592 |
For a decomposition $x = (X_a, M_{ib})$ in $\cell(W)$, define $\psi_\cN(x)$ to be the subset |
108 | 1593 |
\[ |
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|
1594 |
\psi_\cN(x) \sub \left(\prod_a \cC(X_a)\right) \times \left(\prod_{ib} \cN_i(M_{ib})\right) |
108 | 1595 |
\] |
1596 |
such that the restrictions to the various pieces of shared boundaries amongst the |
|
1597 |
$X_a$ and $M_{ib}$ all agree. |
|
435 | 1598 |
(That is, the fibered product over the boundary restriction maps.) |
108 | 1599 |
If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$ |
1600 |
via the gluing (composition or action) maps from $\cC$ and the $\cN_i$. |
|
1601 |
||
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|
1602 |
We now define the set $\cC(W, \cN)$ to be the colimit of the functor $\psi_\cN$. |
435 | 1603 |
(As in \S\ref{ss:ncat-coend}, if $k=n$ we take a colimit in whatever |
1604 |
category we are enriching over, and if additionally we are in the $A_\infty$ case, |
|
1605 |
then we use a homotopy colimit.) |
|
1606 |
||
1607 |
\medskip |
|
108 | 1608 |
|
143 | 1609 |
If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define |
1610 |
$\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold |
|
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1611 |
$D\times Y_i \sub \bd(D\times W)$. |
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|
1612 |
It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$ |
435 | 1613 |
has the structure of an $n{-}k$-category. |
144 | 1614 |
|
1615 |
\medskip |
|
1616 |
||
1617 |
We will use a simple special case of the above |
|
1618 |
construction to define tensor products |
|
1619 |
of modules. |
|
1620 |
Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$. |
|
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|
1621 |
(If $k=1$ and our manifolds are oriented, then one should be |
144 | 1622 |
a left module and the other a right module.) |
1623 |
Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$. |
|
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|
1624 |
Define the tensor product $\cM_1 \tensor \cM_2$ to be the |
435 | 1625 |
$n{-}1$-category associated as above to $J$ with its boundary labeled by $\cM_1$ and $\cM_2$. |
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|
1626 |
This of course depends (functorially) |
144 | 1627 |
on the choice of 1-ball $J$. |
105 | 1628 |
|
144 | 1629 |
We will define a more general self tensor product (categorified coend) below. |
1630 |
||
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1631 |
|
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|
1632 |
|
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|
1633 |
|
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|
1634 |
\subsection{Morphisms of modules} |
288 | 1635 |
\label{ss:module-morphisms} |
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1636 |
|
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|
1637 |
Modules are collections of functors together with some additional data, so we define morphisms |
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|
1638 |
of modules to be collections of natural transformations which are compatible with this |
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|
1639 |
additional data. |
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|
1640 |
|
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|
1641 |
More specifically, let $\cX$ and $\cY$ be $\cC$ modules, i.e.\ collections of functors |
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|
1642 |
$\{\cX_k\}$ and $\{\cY_k\}$, for $0\le k\le n$, from marked $k$-balls to sets |
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|
1643 |
as in Module Axiom \ref{module-axiom-funct}. |
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|
1644 |
A morphism $g:\cX\to\cY$ is a collection of natural transformations $g_k:\cX_k\to\cY_k$ |
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|
1645 |
satisfying: |
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|
1646 |
\begin{itemize} |
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|
1647 |
\item Each $g_k$ commutes with $\bd$. |
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|
1648 |
\item Each $g_k$ commutes with gluing (module composition and $\cC$ action). |
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|
1649 |
\item Each $g_k$ commutes with taking products. |
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|
1650 |
\item In the top dimension $k=n$, $g_n$ preserves whatever additional structure we are enriching over (e.g.\ vector |
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|
1651 |
spaces). |
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diff
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|
1652 |
In the $A_\infty$ case (e.g.\ enriching over chain complexes) $g_n$ should live in |
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|
1653 |
an appropriate derived hom space, as described below. |
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|
1654 |
\end{itemize} |
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|
1655 |
|
546
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|
1656 |
We will be mainly interested in the case $n=1$ and enriched over chain complexes, |
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|
1657 |
since this is the case that's relevant to the generalized Deligne conjecture of \S\ref{sec:deligne}. |
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|
1658 |
So we treat this case in more detail. |
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|
1659 |
|
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|
1660 |
First we explain the remark about derived hom above. |
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|
1661 |
Let $L$ be a marked 1-ball and let $\cl{\cX}(L)$ denote the local homotopy colimit construction |
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|
1662 |
associated to $L$ by $\cX$ and $\cC$. |
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|
1663 |
(See \S \ref{ss:ncat_fields} and \S \ref{moddecss}.) |
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|
1664 |
Define $\cl{\cY}(L)$ similarly. |
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|
1665 |
For $K$ an unmarked 1-ball let $\cl{\cC(K)}$ denote the local homotopy colimit |
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|
1666 |
construction associated to $K$ by $\cC$. |
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|
1667 |
Then we have an injective gluing map |
261
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|
1668 |
\[ |
546
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|
1669 |
\gl: \cl{\cX}(L) \ot \cl{\cC}(K) \to \cl{\cX}(L\cup K) |
261
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|
1670 |
\] |
546
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|
1671 |
which is also a chain map. |
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|
1672 |
(For simplicity we are suppressing mention of boundary conditions on the unmarked |
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|
1673 |
boundary components of the 1-balls.) |
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|
1674 |
We define $\hom_\cC(\cX \to \cY)$ to be a collection of (graded linear) natural transformations |
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|
1675 |
$g: \cl{\cX}(L)\to \cl{\cY}(L)$ such that the following diagram commutes for all $L$ and $K$: |
262
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|
1676 |
\[ \xymatrix{ |
546
689ef4edbdd7
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|
1677 |
\cl{\cX}(L) \ot \cl{\cC}(K) \ar[r]^{\gl} \ar[d]_{g\ot \id} & \cl{\cX}(L\cup K) \ar[d]^{g}\\ |
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|
1678 |
\cl{\cY}(L) \ot \cl{\cC}(K) \ar[r]^{\gl} & \cl{\cY}(L\cup K) |
262
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|
1679 |
} \] |
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|
1680 |
|
546
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|
1681 |
The usual differential on graded linear maps between chain complexes induces a differential |
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|
1682 |
on $\hom_\cC(\cX \to \cY)$, giving it the structure of a chain complex. |
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|
1683 |
|
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|
1684 |
Let $\cZ$ be another $\cC$ module. |
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|
1685 |
We define a chain map |
262
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|
1686 |
\[ |
546
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|
1687 |
a: \hom_\cC(\cX \to \cY) \ot (\cX \ot_\cC \cZ) \to \cY \ot_\cC \cZ |
386 | 1688 |
\] |
546
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|
1689 |
as follows. |
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|
1690 |
Recall that the tensor product $\cX \ot_\cC \cZ$ depends on a choice of interval $J$, labeled |
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|
1691 |
by $\cX$ on one boundary component and $\cZ$ on the other. |
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|
1692 |
Because we are using the {\it local} homotopy colimit, any generator |
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|
1693 |
$D\ot x\ot \bar{c}\ot z$ of $\cX \ot_\cC \cZ$ can be written (perhaps non-uniquely) as a gluing |
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|
1694 |
$(D'\ot x \ot \bar{c}') \bullet (D''\ot \bar{c}''\ot z)$, for some decomposition $J = L'\cup L''$ |
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|
1695 |
and with $D'\ot x \ot \bar{c}'$ a generator of $\cl{\cX}(L')$ and |
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|
1696 |
$D''\ot \bar{c}''\ot z$ a generator of $\cl{\cZ}(L'')$. |
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|
1697 |
(Such a splitting exists because the blob diagram $D$ can be split into left and right halves, |
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|
1698 |
since no blob can include both the leftmost and rightmost intervals in the underlying decomposition. |
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|
1699 |
This step would fail if we were using the usual hocolimit instead of the local hocolimit.) |
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|
1700 |
We now define |
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1701 |
\[ |
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1702 |
a: g\ot (D\ot x\ot \bar{c}\ot z) \mapsto g(D'\ot x \ot \bar{c}')\bullet (D''\ot \bar{c}''\ot z) . |
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1703 |
\] |
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1704 |
This does not depend on the choice of splitting $D = D'\bullet D''$ because $g$ commutes with gluing. |
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1705 |
|
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1706 |
|
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1707 |
|
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1708 |
|
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1709 |
\subsection{The \texorpdfstring{$n{+}1$}{n+1}-category of sphere modules} |
218 | 1710 |
\label{ssec:spherecat} |
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1711 |
|
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1712 |
In this subsection we define $n{+}1$-categories $\cS$ of ``sphere modules" |
327 | 1713 |
whose objects are $n$-categories. |
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1714 |
With future applications in mind, we treat simultaneously the big category |
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1715 |
of all $n$-categories and all sphere modules and also subcategories thereof. |
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1716 |
When $n=1$ this is closely related to familiar $2$-categories consisting of |
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1717 |
algebras, bimodules and intertwiners (or a subcategory of that). |
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1718 |
|
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1719 |
While it is appropriate to call an $S^0$ module a bimodule, |
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|
1720 |
this is much less true for higher dimensional spheres, |
327 | 1721 |
so we prefer the term ``sphere module" for the general case. |
144 | 1722 |
|
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1723 |
%The results of this subsection are not needed for the rest of the paper, |
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1724 |
%so we will skimp on details in a couple of places. We have included this mostly |
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|
1725 |
%for the sake of comparing our notion of a disk-like $n$-category to other definitions. |
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1726 |
|
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1727 |
For simplicity, we will assume that $n$-categories are enriched over $\c$-vector spaces. |
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1728 |
|
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1729 |
The $0$- through $n$-dimensional parts of $\cS$ are various sorts of modules, and we describe |
205 | 1730 |
these first. |
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1731 |
The $n{+}1$-dimensional part of $\cS$ consists of intertwiners |
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1732 |
of $1$-category modules associated to decorated $n$-balls. |
205 | 1733 |
We will see below that in order for these $n{+}1$-morphisms to satisfy all of |
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1734 |
the axioms of an $n{+}1$-category (in particular, duality requirements), we will have to assume |
205 | 1735 |
that our $n$-categories and modules have non-degenerate inner products. |
1736 |
(In other words, we need to assume some extra duality on the $n$-categories and modules.) |
|
1737 |
||
1738 |
\medskip |
|
1739 |
||
1740 |
Our first task is to define an $n$-category $m$-sphere module, for $0\le m \le n-1$. |
|
1741 |
These will be defined in terms of certain classes of marked balls, very similarly |
|
1742 |
to the definition of $n$-category modules above. |
|
1743 |
(This, in turn, is very similar to our definition of $n$-category.) |
|
1744 |
Because of this similarity, we only sketch the definitions below. |
|
1745 |
||
327 | 1746 |
We start with $0$-sphere modules, which also could reasonably be called (categorified) bimodules. |
205 | 1747 |
(For $n=1$ they are precisely bimodules in the usual, uncategorified sense.) |
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1748 |
We prefer the more awkward term ``0-sphere module" to emphasize the analogy |
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1749 |
with the higher sphere modules defined below. |
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1750 |
|
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1751 |
Define a $0$-marked $k$-ball, $1\le k \le n$, to be a pair $(X, M)$ homeomorphic to the standard |
327 | 1752 |
$(B^k, B^{k-1})$. |
209 | 1753 |
See Figure \ref{feb21a}. |
205 | 1754 |
Another way to say this is that $(X, M)$ is homeomorphic to $B^{k-1}\times([-1,1], \{0\})$. |
1755 |
||
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1756 |
\begin{figure}[t] |
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1757 |
$$\tikz[baseline,line width=2pt]{\draw[blue] (-2,0)--(2,0); \fill[red] (0,0) circle (0.1);} \qquad \qquad \tikz[baseline,line width=2pt]{\draw[blue][fill=blue!30!white] (0,0) circle (2 and 1); \draw[red] (0,1)--(0,-1);}$$ |
209 | 1758 |
\caption{0-marked 1-ball and 0-marked 2-ball} |
1759 |
\label{feb21a} |
|
1760 |
\end{figure} |
|
1761 |
||
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1762 |
The $0$-marked balls can be cut into smaller balls in various ways. |
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1763 |
We only consider those decompositions in which the smaller balls are either |
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1764 |
$0$-marked (i.e. intersect the $0$-marking of the large ball in a disc) |
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1765 |
or plain (don't intersect the $0$-marking of the large ball). |
327 | 1766 |
We can also take the boundary of a $0$-marked ball, which is $0$-marked sphere. |
205 | 1767 |
|
1768 |
Fix $n$-categories $\cA$ and $\cB$. |
|
327 | 1769 |
These will label the two halves of a $0$-marked $k$-ball. |
205 | 1770 |
|
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|
1771 |
An $n$-category $0$-sphere module $\cM$ over the $n$-categories $\cA$ and $\cB$ is a collection of functors $\cM_k$ from the category |
327 | 1772 |
of $0$-marked $k$-balls, $1\le k \le n$, |
205 | 1773 |
(with the two halves labeled by $\cA$ and $\cB$) to the category of sets. |
1774 |
If $k=n$ these sets should be enriched to the extent $\cA$ and $\cB$ are. |
|
327 | 1775 |
Given a decomposition of a $0$-marked $k$-ball $X$ into smaller balls $X_i$, we have |
205 | 1776 |
morphism sets $\cA_k(X_i)$ (if $X_i$ lies on the $\cA$-labeled side) |
1777 |
or $\cB_k(X_i)$ (if $X_i$ lies on the $\cB$-labeled side) |
|
1778 |
or $\cM_k(X_i)$ (if $X_i$ intersects the marking and is therefore a smaller 0-marked ball). |
|
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|
1779 |
Corresponding to this decomposition we have a composition (or ``gluing") map |
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|
1780 |
from the product (fibered over the boundary data) of these various sets into $\cM_k(X)$. |
205 | 1781 |
|
1782 |
\medskip |
|
107 | 1783 |
|
327 | 1784 |
Part of the structure of an $n$-category 0-sphere module $\cM$ is captured by saying it is |
206 | 1785 |
a collection $\cD^{ab}$ of $n{-}1$-categories, indexed by pairs $(a, b)$ of objects (0-morphisms) |
1786 |
of $\cA$ and $\cB$. |
|
1787 |
Let $J$ be some standard 0-marked 1-ball (i.e.\ an interval with a marked point in its interior). |
|
1788 |
Given a $j$-ball $X$, $0\le j\le n-1$, we define |
|
1789 |
\[ |
|
1790 |
\cD(X) \deq \cM(X\times J) . |
|
1791 |
\] |
|
1792 |
The product is pinched over the boundary of $J$. |
|
327 | 1793 |
The set $\cD$ breaks into ``blocks" according to the restrictions to the pinched points of $X\times J$ |
209 | 1794 |
(see Figure \ref{feb21b}). |
206 | 1795 |
These restrictions are 0-morphisms $(a, b)$ of $\cA$ and $\cB$. |
107 | 1796 |
|
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1797 |
\begin{figure}[t] \centering |
367 | 1798 |
\begin{tikzpicture}[blue,line width=2pt] |
1799 |
\draw (0,1) -- (0,-1) node[below] {$X$}; |
|
1800 |
||
1801 |
\draw (2,0) -- (4,0) node[below] {$J$}; |
|
1802 |
\fill[red] (3,0) circle (0.1); |
|
1803 |
||
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1804 |
\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 | 1805 |
\draw[red] (top.center) -- (bottom.center); |
1806 |
\fill (a) circle (0.1) node[left] {\color{green!50!brown} $a$}; |
|
1807 |
\fill (b) circle (0.1) node[right] {\color{green!50!brown} $b$}; |
|
1808 |
||
1809 |
\path (bottom) node[below]{$X \times J$}; |
|
1810 |
||
1811 |
\end{tikzpicture} |
|
209 | 1812 |
\caption{The pinched product $X\times J$} |
1813 |
\label{feb21b} |
|
1814 |
\end{figure} |
|
1815 |
||
206 | 1816 |
More generally, consider an interval with interior marked points, and with the complements |
1817 |
of these points labeled by $n$-categories $\cA_i$ ($0\le i\le l$) and the marked points labeled |
|
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|
1818 |
by $\cA_i$-$\cA_{i+1}$ 0-sphere modules $\cM_i$. |
209 | 1819 |
(See Figure \ref{feb21c}.) |
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1820 |
To this data we can apply the coend construction as in \S\ref{moddecss} above |
327 | 1821 |
to obtain an $\cA_0$-$\cA_l$ $0$-sphere module and, forgetfully, an $n{-}1$-category. |
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1822 |
This amounts to a definition of taking tensor products of $0$-sphere modules over $n$-categories. |
205 | 1823 |
|
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1824 |
\begin{figure}[t] \centering |
367 | 1825 |
\begin{tikzpicture}[baseline,line width = 2pt] |
1826 |
\draw[blue] (0,0) -- (6,0); |
|
1827 |
\foreach \x/\n in {0.5/0,1.5/1,3/2,4.5/3,5.5/4} { |
|
1828 |
\path (\x,0) node[below] {\color{green!50!brown}$\cA_{\n}$}; |
|
1829 |
} |
|
1830 |
\foreach \x/\n in {1/0,2/1,4/2,5/3} { |
|
1831 |
\fill[red] (\x,0) circle (0.1) node[above] {\color{green!50!brown}$\cM_{\n}$}; |
|
1832 |
} |
|
1833 |
\end{tikzpicture} |
|
1834 |
\qquad |
|
1835 |
\qquad |
|
1836 |
\begin{tikzpicture}[baseline,line width = 2pt] |
|
1837 |
\draw[blue] (0,0) circle (2); |
|
1838 |
\foreach \q/\n in {-45/0,90/1,180/2} { |
|
1839 |
\path (\q:2.4) node {\color{green!50!brown}$\cA_{\n}$}; |
|
1840 |
} |
|
1841 |
\foreach \q/\n in {60/0,120/1,-120/2} { |
|
1842 |
\fill[red] (\q:2) circle (0.1); |
|
1843 |
\path (\q:2.4) node {\color{green!50!brown}$\cM_{\n}$}; |
|
1844 |
} |
|
1845 |
\end{tikzpicture} |
|
209 | 1846 |
\caption{Marked and labeled 1-manifolds} |
1847 |
\label{feb21c} |
|
1848 |
\end{figure} |
|
1849 |
||
206 | 1850 |
We could also similarly mark and label a circle, obtaining an $n{-}1$-category |
1851 |
associated to the marked and labeled circle. |
|
209 | 1852 |
(See Figure \ref{feb21c}.) |
206 | 1853 |
If the circle is divided into two intervals, we can think of this $n{-}1$-category |
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1854 |
as the 2-sided tensor product of the two 0-sphere modules associated to the two intervals. |
206 | 1855 |
|
1856 |
\medskip |
|
1857 |
||
1858 |
Next we define $n$-category 1-sphere modules. |
|
1859 |
These are just representations of (modules for) $n{-}1$-categories associated to marked and labeled |
|
1860 |
circles (1-spheres) which we just introduced. |
|
1861 |
||
1862 |
Equivalently, we can define 1-sphere modules in terms of 1-marked $k$-balls, $2\le k\le n$. |
|
1863 |
Fix a marked (and labeled) circle $S$. |
|
209 | 1864 |
Let $C(S)$ denote the cone of $S$, a marked 2-ball (Figure \ref{feb21d}). |
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1865 |
%\nn{I need to make up my mind whether marked things are always labeled too. |
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1866 |
%For the time being, let's say they are.} |
207 | 1867 |
A 1-marked $k$-ball is anything homeomorphic to $B^j \times C(S)$, $0\le j\le n-2$, |
1868 |
where $B^j$ is the standard $j$-ball. |
|
399 | 1869 |
A 1-marked $k$-ball can be decomposed in various ways into smaller balls, which are either |
1870 |
(a) smaller 1-marked $k$-balls, (b) 0-marked $k$-balls, or (c) plain $k$-balls. |
|
560 | 1871 |
(See Figure \ref{subdividing1marked}.) |
207 | 1872 |
We now proceed as in the above module definitions. |
1873 |
||
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|
1874 |
\begin{figure}[t] \centering |
367 | 1875 |
\begin{tikzpicture}[baseline,line width = 2pt] |
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1876 |
\draw[blue][fill=blue!15!white] (0,0) circle (2); |
367 | 1877 |
\fill[red] (0,0) circle (0.1); |
1878 |
\foreach \qm/\qa/\n in {70/-30/0, 120/95/1, -120/180/2} { |
|
1879 |
\draw[red] (0,0) -- (\qm:2); |
|
1880 |
\path (\qa:1) node {\color{green!50!brown} $\cA_\n$}; |
|
1881 |
\path (\qm+20:2.5) node(M\n) {\color{green!50!brown} $\cM_\n$}; |
|
1882 |
\draw[line width=1pt, green!50!brown, ->] (M\n.\qm+135) to[out=\qm+135,in=\qm+90] (\qm+5:1.3); |
|
1883 |
} |
|
1884 |
\end{tikzpicture} |
|
557 | 1885 |
\caption{Cone on a marked circle, the prototypical 1-marked ball} |
209 | 1886 |
\label{feb21d} |
1887 |
\end{figure} |
|
1888 |
||
560 | 1889 |
\begin{figure}[t] \centering |
1890 |
\begin{tikzpicture}[baseline,line width = 2pt] |
|
1891 |
\draw[blue][fill=blue!15!white] (0,0) circle (2); |
|
1892 |
\fill[red] (0,0) circle (0.1); |
|
1893 |
\foreach \qm/\qa/\n in {70/-30/0, 120/95/1, -120/180/2} { |
|
1894 |
\draw[red] (0,0) -- (\qm:2); |
|
1895 |
% \path (\qa:1) node {\color{green!50!brown} $\cA_\n$}; |
|
1896 |
% \path (\qm+20:2.5) node(M\n) {\color{green!50!brown} $\cM_\n$}; |
|
1897 |
% \draw[line width=1pt, green!50!brown, ->] (M\n.\qm+135) to[out=\qm+135,in=\qm+90] (\qm+5:1.3); |
|
1898 |
} |
|
1899 |
||
1900 |
||
1901 |
\begin{scope}[black, thin] |
|
1902 |
\clip (0,0) circle (2); |
|
1903 |
\draw (0:1) -- (90:1) -- (180:1) -- (270:1) -- cycle; |
|
1904 |
\draw (90:1) -- (90:2.1); |
|
1905 |
\draw (180:1) -- (180:2.1); |
|
1906 |
\draw (270:1) -- (270:2.1); |
|
1907 |
\draw (0:1) -- (15:2.1); |
|
1908 |
\draw (0:1) -- (315:1.5) -- (270:1); |
|
1909 |
\draw (315:1.5) -- (315:2.1); |
|
1910 |
\end{scope} |
|
1911 |
||
1912 |
\node(0marked) at (2.5,2.25) {$0$-marked ball}; |
|
1913 |
\node(1marked) at (3.5,1) {$1$-marked ball}; |
|
1914 |
\node(plain) at (3,-1) {plain ball}; |
|
1915 |
\draw[line width=1pt, green!50!brown, ->] (0marked.270) to[out=270,in=45] (50:1.1); |
|
1916 |
\draw[line width=1pt, green!50!brown, ->] (1marked.225) to[out=270,in=45] (0.4,0.1); |
|
1917 |
\draw[line width=1pt, green!50!brown, ->] (plain.90) to[out=135,in=45] (-45:1); |
|
1918 |
||
1919 |
\end{tikzpicture} |
|
1920 |
\caption{Subdividing a $1$-marked ball into plain, $0$-marked and $1$-marked balls.} |
|
1921 |
\label{subdividing1marked} |
|
1922 |
\end{figure} |
|
1923 |
||
207 | 1924 |
A $n$-category 1-sphere module is, among other things, an $n{-}2$-category $\cD$ with |
1925 |
\[ |
|
1926 |
\cD(X) \deq \cM(X\times C(S)) . |
|
1927 |
\] |
|
1928 |
The product is pinched over the boundary of $C(S)$. |
|
1929 |
$\cD$ breaks into ``blocks" according to the restriction to the |
|
1930 |
image of $\bd C(S) = S$ in $X\times C(S)$. |
|
1931 |
||
1932 |
More generally, consider a 2-manifold $Y$ |
|
1933 |
(e.g.\ 2-ball or 2-sphere) marked by an embedded 1-complex $K$. |
|
1934 |
The components of $Y\setminus K$ are labeled by $n$-categories, |
|
1935 |
the edges of $K$ are labeled by 0-sphere modules, |
|
1936 |
and the 0-cells of $K$ are labeled by 1-sphere modules. |
|
1937 |
We can now apply the coend construction and obtain an $n{-}2$-category. |
|
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If $Y$ has boundary then this $n{-}2$-category is a module for the $n{-}1$-category |
207 | 1939 |
associated to the (marked, labeled) boundary of $Y$. |
1940 |
In particular, if $\bd Y$ is a 1-sphere then we get a 1-sphere module as defined above. |
|
1941 |
||
1942 |
\medskip |
|
1943 |
||
1944 |
It should now be clear how to define $n$-category $m$-sphere modules for $0\le m \le n-1$. |
|
1945 |
For example, there is an $n{-}2$-category associated to a marked, labeled 2-sphere, |
|
208 | 1946 |
and a 2-sphere module is a representation of such an $n{-}2$-category. |
207 | 1947 |
|
1948 |
\medskip |
|
1949 |
||
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1950 |
We can now define the $n$-or-less-dimensional part of our $n{+}1$-category $\cS$. |
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1951 |
Choose some collection of $n$-categories, then choose some collections of 0-sphere modules between |
207 | 1952 |
these $n$-categories, then choose some collection of 1-sphere modules for the various |
439
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1953 |
possible marked 1-spheres labeled by the $n$-categories and 0-sphere modules, and so on. |
207 | 1954 |
Let $L_i$ denote the collection of $i{-}1$-sphere modules we have chosen. |
1955 |
(For convenience, we declare a $(-1)$-sphere module to be an $n$-category.) |
|
1956 |
There is a wide range of possibilities. |
|
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1957 |
The set $L_0$ could contain infinitely many $n$-categories or just one. |
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1958 |
For each pair of $n$-categories in $L_0$, $L_1$ could contain no 0-sphere modules at all or |
207 | 1959 |
it could contain several. |
208 | 1960 |
The only requirement is that each $k$-sphere module be a module for a $k$-sphere $n{-}k$-category |
1961 |
constructed out of labels taken from $L_j$ for $j<k$. |
|
1962 |
||
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1963 |
We now define $\cS(X)$, for $X$ a ball of dimension at most $n$, to be the set of all |
208 | 1964 |
cell-complexes $K$ embedded in $X$, with the codimension-$j$ parts of $(X, K)$ labeled |
1965 |
by elements of $L_j$. |
|
1966 |
As described above, we can think of each decorated $k$-ball as defining a $k{-}1$-sphere module |
|
1967 |
for the $n{-}k{+}1$-category associated to its decorated boundary. |
|
1968 |
Thus the $k$-morphisms of $\cS$ (for $k\le n$) can be thought |
|
1969 |
of as $n$-category $k{-}1$-sphere modules |
|
1970 |
(generalizations of bimodules). |
|
387
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1971 |
On the other hand, we can equally well think of the $k$-morphisms as decorations on $k$-balls, |
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1972 |
and from this point of view it is clear that they satisfy all of the axioms of an |
208 | 1973 |
$n{+}1$-category. |
1974 |
(All of the axioms for the less-than-$n{+}1$-dimensional part of an $n{+}1$-category, that is.) |
|
1975 |
||
1976 |
\medskip |
|
1977 |
||
1978 |
Next we define the $n{+}1$-morphisms of $\cS$. |
|
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1979 |
The construction of the 0- through $n$-morphisms was easy and tautological, but the |
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1980 |
$n{+}1$-morphisms will require some effort and combinatorial topology, as well as additional |
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1981 |
duality assumptions on the lower morphisms. These are required because we define the spaces of $n{+}1$-morphisms by making arbitrary choices of incoming and outgoing boundaries for each $n$-ball. The additional duality assumptions are needed to prove independence of our definition form these choices. |
208 | 1982 |
|
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1983 |
Let $X$ be an $n{+}1$-ball, and let $c$ be a decoration of its boundary |
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1984 |
by a cell complex labeled by 0- through $n$-morphisms, as above. |
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1985 |
Choose an $n{-}1$-sphere $E\sub \bd X$ which divides |
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1986 |
$\bd X$ into ``incoming" and ``outgoing" boundary $\bd_-X$ and $\bd_+X$. |
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1987 |
Let $E_c$ denote $E$ decorated by the restriction of $c$ to $E$. |
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1988 |
Recall from above the associated 1-category $\cS(E_c)$. |
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1989 |
We can also have $\cS(E_c)$ modules $\cS(\bd_-X_c)$ and $\cS(\bd_+X_c)$. |
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1990 |
Define |
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1991 |
\[ |
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1992 |
\cS(X; c; E) \deq \hom_{\cS(E_c)}(\cS(\bd_-X_c), \cS(\bd_+X_c)) . |
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1993 |
\] |
208 | 1994 |
|
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1995 |
We will show that if the sphere modules are equipped with a ``compatible family of |
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1996 |
non-degenerate inner products", then there is a coherent family of isomorphisms |
387
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1997 |
$\cS(X; c; E) \cong \cS(X; c; E')$ for all pairs of choices $E$ and $E'$. |
439
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1998 |
This will allow us to define $\cS(X; c)$ independently of the choice of $E$. |
208 | 1999 |
|
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2000 |
First we must define ``inner product", ``non-degenerate" and ``compatible". |
387
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2001 |
Let $Y$ be a decorated $n$-ball, and $\ol{Y}$ it's mirror image. |
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2002 |
(We assume we are working in the unoriented category.) |
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2003 |
Let $Y\cup\ol{Y}$ denote the decorated $n$-sphere obtained by gluing $Y$ and $\ol{Y}$ |
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2004 |
along their common boundary. |
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2005 |
An {\it inner product} on $\cS(Y)$ is a dual vector |
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2006 |
\[ |
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2007 |
z_Y : \cS(Y\cup\ol{Y}) \to \c. |
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2008 |
\] |
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2009 |
We will also use the notation |
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2010 |
\[ |
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2011 |
\langle a, b\rangle \deq z_Y(a\bullet \ol{b}) \in \c . |
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2012 |
\] |
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2013 |
An inner product induces a linear map |
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2014 |
\begin{eqnarray*} |
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2015 |
\varphi: \cS(Y) &\to& \cS(Y)^* \\ |
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2016 |
a &\mapsto& \langle a, \cdot \rangle |
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2017 |
\end{eqnarray*} |
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2018 |
which satisfies, for all morphisms $e$ of $\cS(\bd Y)$, |
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2019 |
\[ |
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2020 |
\varphi(ae)(b) = \langle ae, b \rangle = z_Y(a\bullet e\bullet b) = |
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2021 |
\langle a, eb \rangle = \varphi(a)(eb) . |
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2022 |
\] |
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2023 |
In other words, $\varphi$ is a map of $\cS(\bd Y)$ modules. |
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2024 |
An inner product is {\it non-degenerate} if $\varphi$ is an isomorphism. |
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2025 |
This implies that $\cS(Y; c)$ is finite dimensional for all boundary conditions $c$. |
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2026 |
(One can think of these inner products as giving some duality in dimension $n{+}1$; |
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2027 |
heretofore we have only assumed duality in dimensions 0 through $n$.) |
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2028 |
|
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|
2029 |
Next we define compatibility. |
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2030 |
Let $Y = Y_1\cup Y_2$ with $D = Y_1\cap Y_2$. |
398
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2031 |
Let $X_1$ and $X_2$ be the two components of $Y\times I$ cut along |
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2032 |
$D\times I$, in both cases using the pinched product. |
390
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2033 |
(Here we are overloading notation and letting $D$ denote both a decorated and an undecorated |
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2034 |
manifold.) |
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2035 |
We have $\bd X_i = Y_i \cup \ol{Y}_i \cup (D\times I)$ |
393 | 2036 |
(see Figure \ref{jun23a}). |
2037 |
\begin{figure}[t] |
|
2038 |
\begin{equation*} |
|
497 | 2039 |
\mathfig{.6}{ncat/YxI-sliced} |
393 | 2040 |
\end{equation*} |
2041 |
\caption{$Y\times I$ sliced open} |
|
2042 |
\label{jun23a} |
|
2043 |
\end{figure} |
|
390
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2044 |
Given $a_i\in \cS(Y_i)$, $b_i\in \cS(\ol{Y}_i)$ and $v\in\cS(D\times I)$ |
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2045 |
which agree on their boundaries, we can evaluate |
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2046 |
\[ |
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2047 |
z_{Y_i}(a_i\bullet b_i\bullet v) \in \c . |
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2048 |
\] |
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2049 |
(This requires a choice of homeomorphism $Y_i \cup \ol{Y}_i \cup (D\times I) \cong |
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2050 |
Y_i \cup \ol{Y}_i$, but the value of $z_{Y_i}$ is independent of this choice.) |
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2051 |
We can think of $z_{Y_i}$ as giving a function |
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2052 |
\[ |
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2053 |
\psi_i : \cS(Y_i) \ot \cS(\ol{Y}_i) \to \cS(D\times I)^* |
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2054 |
\stackrel{\varphi\inv}{\longrightarrow} \cS(D\times I) . |
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2055 |
\] |
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2056 |
We can now finally define a family of inner products to be {\it compatible} if |
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2057 |
for all decompositions $Y = Y_1\cup Y_2$ as above and all $a_i\in \cS(Y_i)$, $b_i\in \cS(\ol{Y}_i)$ |
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2058 |
we have |
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2059 |
\[ |
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2060 |
z_Y(a_1\bullet a_2\bullet b_1\bullet b_2) = |
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2061 |
z_{D\times I}(\psi_1(a_1\ot b_1)\bullet \psi_2(a_2\ot b_2)) . |
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2062 |
\] |
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2063 |
In other words, the inner product on $Y$ is determined by the inner products on |
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2064 |
$Y_1$, $Y_2$ and $D\times I$. |
207 | 2065 |
|
392
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2066 |
Now we show how to unambiguously identify $\cS(X; c; E)$ and $\cS(X; c; E')$ for any |
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2067 |
two choices of $E$ and $E'$. |
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|
2068 |
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
|
2069 |
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
|
2070 |
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
|
2071 |
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
|
2072 |
Let $D = B\cap A$. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2073 |
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
|
2074 |
\[ |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2075 |
\psi: \cS(B)\ot\cS(\ol{B}) \to \cS(D\times I) . |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2076 |
\] |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2077 |
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
|
2078 |
to be the composition |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2079 |
\[ |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2080 |
\cS(C\cup \ol{B}) \stackrel{f\ot\id}{\longrightarrow} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2081 |
\cS(A\cup B\cup \ol{B}) \stackrel{\id\ot\psi}{\longrightarrow} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2082 |
\cS(A\cup(D\times I)) \stackrel{\cong}{\longrightarrow} \cS(A) . |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2083 |
\] |
393 | 2084 |
(See Figure \ref{jun23b}.) |
2085 |
\begin{figure}[t] |
|
443 | 2086 |
$$ |
2087 |
\begin{tikzpicture}[baseline,line width = 1pt,x=1.5cm,y=1.5cm] |
|
2088 |
\draw (0,0) node(R) {} |
|
2089 |
-- (0.75,0) node[below] {$\bar{B}$} |
|
2090 |
--(1.5,0) node[circle,fill=black,inner sep=2pt] {} |
|
2091 |
arc (0:80:1.5) node[above] {$D \times I$} |
|
2092 |
arc (80:180:1.5); |
|
2093 |
\foreach \r in {0.3, 0.6, 0.9, 1.2} { |
|
2094 |
\draw[blue!50, line width = 0.5pt] (\r,0) arc (0:180:\r); |
|
2095 |
} |
|
2096 |
\draw[fill=white] |
|
2097 |
(R) node[circle,fill=black,inner sep=2pt] {} |
|
2098 |
arc (45:65:3) node[below] {$B$} |
|
2099 |
arc (65:90:3) node[below] {$A$} |
|
2100 |
arc (90:135:3) node[circle,fill=black,inner sep=2pt] {} |
|
2101 |
arc (-135:-90:3) node[below] {$C$} |
|
2102 |
arc (-90:-45:3); |
|
2103 |
\draw[fill] (150:1.5) circle (2pt) node[above=4pt] {$D$}; |
|
547
fbad527790c1
minor: futzing with font size in 2 figs
Kevin Walker <kevin@canyon23.net>
parents:
546
diff
changeset
|
2104 |
\node[green!50!brown] at (-2,0) {\scalebox{1.4}{$\uparrow f$}}; |
fbad527790c1
minor: futzing with font size in 2 figs
Kevin Walker <kevin@canyon23.net>
parents:
546
diff
changeset
|
2105 |
\node[green!50!brown] at (0.2,0.8) {\scalebox{1.4}{$\uparrow \psi$}}; |
443 | 2106 |
\end{tikzpicture} |
2107 |
$$ |
|
393 | 2108 |
\caption{Moving $B$ from top to bottom} |
2109 |
\label{jun23b} |
|
2110 |
\end{figure} |
|
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2111 |
Let $D' = B\cap C$. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2112 |
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
|
2113 |
\[ |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2114 |
\psi^\dagger: \cS(D'\times I) \to \cS(\ol{B})\ot\cS(B) . |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2115 |
\] |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2116 |
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
|
2117 |
to be the composition |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2118 |
\[ |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2119 |
\cS(C) \stackrel{\cong}{\longrightarrow} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2120 |
\cS(C\cup(D'\times I)) \stackrel{\id\ot\psi^\dagger}{\longrightarrow} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2121 |
\cS(C\cup \ol{B}\cup B) \stackrel{f'\ot\id}{\longrightarrow} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2122 |
\cS(A\cup B) . |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2123 |
\] |
393 | 2124 |
(See Figure \ref{jun23c}.) |
2125 |
\begin{figure}[t] |
|
2126 |
\begin{equation*} |
|
443 | 2127 |
\begin{tikzpicture}[baseline,line width = 1pt,x=1.5cm,y=-1.5cm] |
2128 |
\draw (0,0) node(R) {} |
|
2129 |
-- (0.75,0) node[above] {$B$} |
|
2130 |
--(1.5,0) node[circle,fill=black,inner sep=2pt] {} |
|
2131 |
arc (0:80:1.5) node[below] {$D' \times I$} |
|
2132 |
arc (80:180:1.5); |
|
2133 |
\foreach \r in {0.3, 0.6, 0.9, 1.2} { |
|
2134 |
\draw[blue!50, line width = 0.5pt] (\r,0) arc (0:180:\r); |
|
2135 |
} |
|
2136 |
\draw[fill=white] |
|
2137 |
(R) node[circle,fill=black,inner sep=2pt] {} |
|
2138 |
arc (45:65:3) node[above] {$\bar{B}$} |
|
2139 |
arc (65:90:3) node[below] {$C$} |
|
2140 |
arc (90:135:3) node[circle,fill=black,inner sep=2pt] {} |
|
2141 |
arc (-135:-90:3) node[below] {$A$} |
|
2142 |
arc (-90:-45:3); |
|
2143 |
\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
|
2144 |
\node[green!50!brown] at (-2,0) {\scalebox{1.4}{$f'\uparrow $}}; |
fbad527790c1
minor: futzing with font size in 2 figs
Kevin Walker <kevin@canyon23.net>
parents:
546
diff
changeset
|
2145 |
\node[green!50!brown] at (0.2,0.8) {\scalebox{1.4}{$\psi^\dagger \uparrow $}}; |
443 | 2146 |
\end{tikzpicture} |
393 | 2147 |
\end{equation*} |
2148 |
\caption{Moving $B$ from bottom to top} |
|
2149 |
\label{jun23c} |
|
2150 |
\end{figure} |
|
2151 |
Let $D' = B\cap C$. |
|
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2152 |
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
|
2153 |
|
559 | 2154 |
\begin{lem} \label{equator-lemma} |
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2155 |
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
|
2156 |
\end{lem} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2157 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2158 |
\begin{proof} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2159 |
(Sketch) |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2160 |
$E$ 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
|
2161 |
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
|
2162 |
(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
|
2163 |
\end{proof} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2164 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2165 |
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
|
2166 |
between $\cS(X; c; E)$ and $\cS(X; c; E')$ for any pair $E$, $E'$. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2167 |
This construction involves on a choice of simple ``moves" (as above) to transform |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2168 |
$E$ to $E'$. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2169 |
We must now show that the isomorphism does not depend on this choice. |
505
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2170 |
We will show below that it suffice to check two ``movie moves". |
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2171 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2172 |
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
|
2173 |
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
|
2174 |
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
|
2175 |
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
|
2176 |
|
439
10f0f68cafb4
mostly (entirely?) ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
435
diff
changeset
|
2177 |
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
|
2178 |
across $B_1$ and $B_2$, say, with a single push across $B_1\cup B_2$. |
393 | 2179 |
(See Figure \ref{jun23d}.) |
2180 |
\begin{figure}[t] |
|
456
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2181 |
\begin{tikzpicture} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2182 |
\node(L) { |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2183 |
\scalebox{0.5}{ |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2184 |
\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
|
2185 |
\draw[red] (0.75,0) -- +(2,0); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2186 |
\draw[red] (0,0) node(R) {} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2187 |
-- (0.75,0) node[below] {} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2188 |
--(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
|
2189 |
\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
|
2190 |
\draw (1.5,0) arc (0:149:1.5); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2191 |
\draw[red] |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2192 |
(R) node[circle,fill=black,inner sep=2pt] {} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2193 |
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
|
2194 |
\draw[red] (-5.5,0) -- (-4.2,0); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2195 |
\draw (R) arc (45:75:3); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2196 |
\draw (150:1.5) arc (74:135:3); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2197 |
\node at (-2,0) {\scalebox{2.0}{$B_1$}}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2198 |
\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
|
2199 |
\node at (-4,1.2) {\scalebox{2.0}{$A$}}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2200 |
\node at (-4,-1.2) {\scalebox{2.0}{$C$}}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2201 |
\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
|
2202 |
\end{tikzpicture} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2203 |
} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2204 |
}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2205 |
\node(M) at (5,4) { |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2206 |
\scalebox{0.5}{ |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2207 |
\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
|
2208 |
\draw[red] (0.75,0) -- +(2,0); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2209 |
\draw[red] (0,0) node(R) {} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2210 |
-- (0.75,0) node[below] {} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2211 |
--(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
|
2212 |
\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
|
2213 |
\draw(1.5,0) arc (0:149:1.5); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2214 |
\draw |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2215 |
(R) node[circle,fill=black,inner sep=2pt] {} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2216 |
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
|
2217 |
\draw[red] (-5.5,0) -- (-4.2,0); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2218 |
\draw[red] (R) arc (45:75:3); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2219 |
\draw[red] (150:1.5) arc (74:135:3); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2220 |
\node at (-2,0) {\scalebox{2.0}{$B_1$}}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2221 |
\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
|
2222 |
\node at (-4,1.2) {\scalebox{2.0}{$A$}}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2223 |
\node at (-4,-1.2) {\scalebox{2.0}{$C$}}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2224 |
\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
|
2225 |
\end{tikzpicture} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2226 |
} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2227 |
}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2228 |
\node(R) at (10,0) { |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2229 |
\scalebox{0.5}{ |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2230 |
\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
|
2231 |
\draw[red] (0.75,0) -- +(2,0); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2232 |
\draw (0,0) node(R) {} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2233 |
-- (0.75,0) node[below] {} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2234 |
--(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
|
2235 |
\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
|
2236 |
\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
|
2237 |
\draw |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2238 |
(R) node[circle,fill=black,inner sep=2pt] {} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2239 |
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
|
2240 |
\draw[red] (-5.5,0) -- (-4.2,0); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2241 |
\draw (R) arc (45:75:3); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2242 |
\draw[red] (150:1.5) arc (74:135:3); |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2243 |
\node at (-2,0) {\scalebox{2.0}{$B_1$}}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2244 |
\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
|
2245 |
\node at (-4,1.2) {\scalebox{2.0}{$A$}}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2246 |
\node at (-4,-1.2) {\scalebox{2.0}{$C$}}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2247 |
\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
|
2248 |
\end{tikzpicture} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2249 |
} |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2250 |
}; |
a5d75e0f9229
filtration -> simplex, and another diagram
Scott Morrison <scott@tqft.net>
parents:
448
diff
changeset
|
2251 |
\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
|
2252 |
\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
|
2253 |
\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
|
2254 |
\end{tikzpicture} |
393 | 2255 |
\caption{A movie move} |
2256 |
\label{jun23d} |
|
2257 |
\end{figure} |
|
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2258 |
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
|
2259 |
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
|
2260 |
|
505
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2261 |
%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
|
2262 |
%\nn{...} |
439
10f0f68cafb4
mostly (entirely?) ncat revisions
Kevin Walker <kevin@canyon23.net>
parents:
435
diff
changeset
|
2263 |
|
505
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2264 |
If $n\ge 2$, these two movie move suffice: |
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2265 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2266 |
\begin{lem} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2267 |
Assume $n\ge 2$ and fix $E$ and $E'$ as above. |
550 | 2268 |
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
|
2269 |
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
|
2270 |
\end{lem} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2271 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2272 |
\begin{proof} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2273 |
(Sketch) |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2274 |
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
|
2275 |
of $\bd X$. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2276 |
Up to homotopy, |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2277 |
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
|
2278 |
into small families which are either |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2279 |
(a) supported away from $E$, |
505
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2280 |
(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
|
2281 |
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
|
2282 |
(This fails for $n=1$.) |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2283 |
\end{proof} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2284 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2285 |
For $n=1$ we have to check an additional ``global" relations corresponding to |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2286 |
rotating the 0-sphere $E$ around the 1-sphere $\bd X$. |
529 | 2287 |
But if $n=1$, then we are in the case of ordinary algebroids and bimodules, |
560 | 2288 |
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
|
2289 |
|
505
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2290 |
\medskip |
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2291 |
|
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2292 |
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
|
2293 |
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
|
2294 |
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
|
2295 |
\[ |
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2296 |
\cS(X; c) \cong \cS(X; c; E) \deq \hom_{\cS(E_c)}(\cS(\bd_-X_c), \cS(\bd_+X_c)) . |
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2297 |
\] |
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2298 |
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
|
2299 |
\[ |
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2300 |
f: \cS(X; c; E) \to \cS(X'; f(c); f(E)) . |
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2301 |
\] |
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2302 |
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
|
2303 |
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
|
2304 |
In particular, if $F: X\to X$ is the identity on $\bd X$ then $f$ acts trivially, as required by |
552 | 2305 |
Axiom \ref{axiom:extended-isotopies}. |
505
8ed3aeb78778
sphere module n+1 mor stuff
Kevin Walker <kevin@canyon23.net>
parents:
497
diff
changeset
|
2306 |
|
506 | 2307 |
We define product $n{+}1$-morphisms to be identity maps of modules. |
101 | 2308 |
|
506 | 2309 |
To define (binary) composition of $n{+}1$-morphisms, choose the obvious common equator |
2310 |
then compose the module maps. |
|
559 | 2311 |
The proof that this composition rule is associative is similar to the proof of Lemma \ref{equator-lemma}. |