author | scott@6e1638ff-ae45-0410-89bd-df963105f760 |
Fri, 11 Dec 2009 23:16:03 +0000 | |
changeset 188 | 4f7300effe49 |
parent 187 | 4067c74547bb |
child 189 | a3631a999462 |
permissions | -rw-r--r-- |
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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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\section{$n$-categories} |
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\label{sec:ncats} |
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%In order to make further progress establishing properties of the blob complex, |
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%we need a definition of $A_\infty$ $n$-category that is adapted to our needs. |
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%(Even in the case $n=1$, we need the new definition given below.) |
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%It turns out that the $A_\infty$ $n$-category definition and the plain $n$-category |
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%definition are mostly the same, so we give a new definition of plain |
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%$n$-categories too. |
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%We also define modules and tensor products for both plain and $A_\infty$ $n$-categories. |
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\subsection{Definition of $n$-categories} |
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Before proceeding, we need more appropriate definitions of $n$-categories, |
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$A_\infty$ $n$-categories, modules for these, and tensor products of these modules. |
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(As is the case throughout this paper, by ``$n$-category" we implicitly intend 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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For examples of topological origin, it is typically easy to show that they |
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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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\medskip |
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Consider first ordinary $n$-categories. |
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We need a set (or sets) of $k$-morphisms for each $0\le k \le n$. |
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We must decide on the ``shape" of the $k$-morphisms. |
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Some $n$-category definitions model $k$-morphisms on the standard bihedron (interval, bigon, ...). |
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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.) |
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Still other definitions \nn{need refs for all these; maybe the Leinster book \cite{MR2094071}} |
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model the $k$-morphisms on more complicated combinatorial polyhedra. |
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We will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to |
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the standard $k$-ball. |
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In other words, |
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\begin{preliminary-axiom}{\ref{axiom:morphisms}}{Morphisms} |
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For any $k$-manifold $X$ homeomorphic |
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to the standard $k$-ball, we have a set of $k$-morphisms |
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$\cC_k(X)$. |
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\end{preliminary-axiom} |
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Terminology: 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. |
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The same goes for ``a $k$-sphere" below. |
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Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on |
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the boundary), we want a corresponding |
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bijection of sets $f:\cC(X)\to \cC(Y)$. |
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(This will imply ``strong duality", among other things.) |
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So we replace the above with |
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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 usually omit the subscript $k$.) |
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We are so far being deliberately vague about what flavor of manifolds 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 --- see pseudo-isotopy below} |
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(If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need |
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to be fussier about corners.) |
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For each flavor of manifold there is a corresponding flavor of $n$-category. |
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We will concentrate of the case of PL unoriented manifolds. |
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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-morphsims parameterized |
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by oriented 1-balls.) |
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For $k>1$ and in the presence of strong duality the domain/range division makes less sense. |
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\nn{maybe say more here; rotate disk, Frobenius reciprocity blah blah} |
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We prefer to 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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\begin{axiom}[Boundaries (spheres)] |
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For each $0 \le k \le n-1$, we have a functor $\cC_k$ from |
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the category of $k$-spheres and |
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homeomorphisms to the category of sets and bijections. |
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\end{axiom} |
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(In order to conserve symbols, we use the same symbol $\cC_k$ for both morphisms and boundaries.) |
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\begin{axiom}[Boundaries (maps)] |
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For each $k$-ball $X$, we have a map of sets $\bd: \cC(X)\to \cC(\bd X)$. |
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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 $c\in\cC(\bd(X))$, let $\cC(X; c) \deq \bd^{-1}(c)$. |
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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 $c\in \cC(\bd X)$, have the structure of an object in some auxiliary category |
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(e.g.\ vector spaces, or modules over some ring, or chain complexes), |
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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 $n$; |
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$\cC(Y; c)$ is just a plain set if $\dim(Y) < n$. |
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\medskip |
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\nn{ |
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%At the moment I'm a little confused about orientations, and more specifically |
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%about the role of orientation-reversing maps of boundaries when gluing oriented manifolds. |
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Maybe need a discussion about what the boundary of a manifold with a |
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structure (e.g. orientation) means. |
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Tentatively, I think we need to redefine the oriented boundary of an oriented $n$-manifold. |
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Instead of an ordinary oriented $(n-1)$-manifold via the inward (or outward) normal |
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first (or last) convention, perhaps it is better to define the boundary to be an $(n-1)$-manifold |
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equipped with an orientation of its once-stabilized tangent bundle. |
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Similarly, in dimension $n-k$ we would have manifolds equipped with an orientation of |
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their $k$ times stabilized tangent bundles. |
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(cf. [Stolz and Teichner].) |
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Probably should also have a framing of the stabilized dimensions in order to indicate which |
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side the bounded manifold is on. |
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For the moment just stick with unoriented manifolds.} |
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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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\begin{axiom}[Boundary from domain and range] |
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Let $S = B_1 \cup_E B_2$, where $S$ is a $k$-sphere $(0\le k\le n-1)$, |
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$B_i$ is a $k$-ball, and $E = B_1\cap B_2$ is a $k{-}1$-sphere (Figure \ref{blah3}). |
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Let $\cC(B_1) \times_{\cC(E)} \cC(B_2)$ denote the fibered product of the |
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two maps $\bd: \cC(B_i)\to \cC(E)$. |
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Then we have an injective map |
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\[ |
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\gl_E : \cC(B_1) \times_{\cC(E)} \cC(B_2) \to \cC(S) |
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\] |
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which is natural with respect to the actions of homeomorphisms. |
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\end{axiom} |
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\begin{figure}[!ht] |
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$$ |
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\begin{tikzpicture}[every label/.style={green}] |
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\node[fill=black, circle, label=below:$E$](S) at (0,0) {}; |
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\node[fill=black, circle, label=above:$E$](N) at (0,2) {}; |
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\draw (S) arc (-90:90:1); |
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\draw (N) arc (90:270:1); |
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\node[left] at (-1,1) {$B_1$}; |
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\node[right] at (1,1) {$B_2$}; |
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\end{tikzpicture} |
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$$ |
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$$\mathfig{.4}{tempkw/blah3}$$ |
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\caption{Combining two balls to get a full boundary}\label{blah3}\end{figure} |
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Note that we insist on injectivity above. |
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Let $\cC(S)_E$ denote the image of $\gl_E$. |
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We will refer to elements of $\cC(S)_E$ as ``splittable along $E$" or ``transverse to $E$". |
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We will call the projection $\cC(S)_E \to \cC(B_i)$ |
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a {\it restriction} map and write $\res_{B_i}(a)$ |
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(or simply $\res(a)$ when there is no ambiguity), for $a\in \cC(S)_E$. |
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These restriction maps can be thought of as |
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domain and range maps, relative to the choice of splitting $S = B_1 \cup_E B_2$. |
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If $B$ is a $k$-ball and $E \sub \bd B$ splits $\bd B$ into two $k{-}1$-balls |
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as above, then we define $\cC(B)_E = \bd^{-1}(\cC(\bd B)_E)$. |
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Next we consider composition of morphisms. |
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For $n$-categories which lack strong duality, one usually considers |
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$k$ different types of composition of $k$-morphisms, each associated to a different direction. |
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(For example, vertical and horizontal composition of 2-morphisms.) |
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In the presence of strong duality, these $k$ distinct compositions are subsumed into |
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one general type of composition which can be in any ``direction". |
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\begin{axiom}[Composition] |
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Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
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and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
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Let $E = \bd Y$, which is a $k{-}2$-sphere. |
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Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
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We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$. |
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Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. |
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Then (axiom) we have a map |
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\[ |
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\gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E |
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\] |
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which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
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to the intersection of the boundaries of $B$ and $B_i$. |
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If $k < n$ we require that $\gl_Y$ is injective. |
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(For $k=n$, see below.) |
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\end{axiom} |
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\begin{figure}[!ht] |
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$$\mathfig{.4}{tempkw/blah5}$$ |
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\caption{From two balls to one ball}\label{blah5}\end{figure} |
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\begin{axiom}[Strict associativity] |
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The composition (gluing) maps above are strictly associative. |
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\end{axiom} |
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\begin{figure}[!ht] |
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$$\mathfig{.65}{tempkw/blah6}$$ |
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\caption{An example of strict associativity}\label{blah6}\end{figure} |
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Notation: $a\bullet b \deq \gl_Y(a, b)$ and/or $a\cup b \deq \gl_Y(a, b)$. |
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In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ |
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a {\it restriction} map and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$. |
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Compositions of boundary and restriction maps will also be called restriction maps. |
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For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a |
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restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$. |
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||
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%More notation and terminology: |
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%We will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ a {\it restriction} |
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%map |
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The above two axioms are equivalent to the following axiom, |
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which we state in slightly vague form. |
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\xxpar{Multi-composition:} |
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{Given any decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball |
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into small $k$-balls, there is a |
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map from an appropriate subset (like a fibered product) |
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of $\cC(B_1)\times\cdots\times\cC(B_m)$ to $\cC(B)$, |
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and these various $m$-fold composition maps satisfy an |
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operad-type strict associativity condition (Figure \ref{blah7}).} |
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\begin{figure}[!ht] |
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$$\mathfig{.8}{tempkw/blah7}$$ |
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\caption{Operadish composition and associativity}\label{blah7}\end{figure} |
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The next axiom is related to identity morphisms, though that might not be immediately obvious. |
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\begin{axiom}[Product (identity) morphisms] |
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Let $X$ be a $k$-ball and $D$ be an $m$-ball, with $k+m \le n$. |
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Then we have a map $\cC(X)\to \cC(X\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. |
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If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram |
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\[ \xymatrix{ |
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X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\ |
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X \ar[r]^{f} & X' |
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} \] |
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commutes, then we have |
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\[ |
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\tilde{f}(a\times D) = f(a)\times D' . |
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\] |
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Product morphisms are compatible with gluing (composition) in both factors: |
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\[ |
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(a'\times D)\bullet(a''\times D) = (a'\bullet a'')\times D |
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\] |
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and |
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\[ |
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(a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') . |
|
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\] |
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\nn{if pinched boundary, then remove first case above} |
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Product morphisms are associative: |
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\[ |
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(a\times D)\times D' = a\times (D\times D') . |
|
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\] |
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(Here we are implicitly using functoriality and the obvious homeomorphism |
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$(X\times D)\times D' \to X\times(D\times D')$.) |
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Product morphisms are compatible with restriction: |
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\[ |
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\res_{X\times E}(a\times D) = a\times E |
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\] |
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for $E\sub \bd D$ and $a\in \cC(X)$. |
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\end{axiom} |
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\nn{need even more subaxioms for product morphisms?} |
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\nn{Almost certainly we need a little more than the above axiom. |
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More specifically, in order to bootstrap our way from the top dimension |
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properties of identity morphisms to low dimensions, we need regular products, |
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pinched products and even half-pinched products. |
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I'm not sure what the best way to cleanly axiomatize the properties of these various |
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products is. |
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For the moment, I'll assume that all flavors of the product are at |
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our disposal, and I'll plan on revising the axioms later.} |
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||
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\nn{current idea for fixing this: make the above axiom a ``preliminary version" |
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(as we have already done with some of the other axioms), then state the official |
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axiom for maps $\pi: E \to X$ which are almost fiber bundles. |
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one option is to restrict E to be a (full/half/not)-pinched product (up to homeo). |
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the alternative is to give some sort of local criterion for what's allowed. |
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state a gluing axiom for decomps $E = E'\cup E''$ where all three are of the correct type. |
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} |
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All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |
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The last axiom (below), concerning actions of |
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homeomorphisms in the top dimension $n$, distinguishes the two cases. |
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||
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We start with the plain $n$-category case. |
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||
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\begin{preliminary-axiom}{\ref{axiom:extended-isotopies}}{Isotopy invariance in dimension $n$} |
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Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
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to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
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Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$. |
316 |
\end{preliminary-axiom} |
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|
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This axiom needs to be strengthened to force product morphisms to act as the identity. |
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Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. |
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Let $J$ be a 1-ball (interval). |
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We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$. |
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(Here we use the ``pinched" version of $Y\times J$. |
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\nn{need notation for this}) |
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We define a map |
325 |
\begin{eqnarray*} |
|
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\psi_{Y,J}: \cC(X) &\to& \cC(X) \\ |
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a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) . |
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\end{eqnarray*} |
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(See Figure \ref{glue-collar}.) |
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\begin{figure}[!ht]\begin{equation*} |
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182 | 331 |
\mathfig{.9}{tempkw/blah10} |
142 | 332 |
\end{equation*}\caption{Extended homeomorphism.}\label{glue-collar}\end{figure} |
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We say that $\psi_{Y,J}$ is {\it extended isotopic} to the identity map. |
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\nn{bad terminology; fix it later} |
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\nn{also need to make clear that plain old isotopic to the identity implies |
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extended isotopic} |
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\nn{maybe remark that in some examples (e.g.\ ones based on sub cell complexes) |
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extended isotopies are also plain isotopies, so |
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no extension necessary} |
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It can be thought of as the action of the inverse of |
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a map which projects a collar neighborhood of $Y$ onto $Y$. |
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||
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The revised axiom is |
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||
187 | 345 |
\begin{axiom}[Extended isotopy invariance in dimension $n$] |
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\label{axiom:extended-isotopies} |
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Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
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to the identity on $\bd X$ and is extended isotopic (rel boundary) to the identity. |
187 | 349 |
Then $f$ acts trivially on $\cC(X)$. |
350 |
\end{axiom} |
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96 | 351 |
|
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\nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
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97 | 354 |
\smallskip |
355 |
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356 |
For $A_\infty$ $n$-categories, we replace |
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isotopy invariance with the requirement that families of homeomorphisms act. |
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For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
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187 | 360 |
\begin{axiom}[Families of homeomorphisms act in dimension $n$] |
361 |
For each $n$-ball $X$ and each $c\in \cC(\bd X)$ we have a map of chain complexes |
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97 | 362 |
\[ |
363 |
C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
|
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\] |
|
365 |
Here $C_*$ means singular chains and $\Homeo_\bd(X)$ is the space of homeomorphisms of $X$ |
|
366 |
which fix $\bd X$. |
|
367 |
These action maps are required to be associative up to homotopy |
|
368 |
\nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
|
369 |
a diagram like the one in Proposition \ref{CDprop} commutes. |
|
370 |
\nn{repeat diagram here?} |
|
187 | 371 |
\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} |
372 |
\end{axiom} |
|
97 | 373 |
|
374 |
We should strengthen the above axiom to apply to families of extended homeomorphisms. |
|
109 | 375 |
To do this we need to explain how extended homeomorphisms form a topological space. |
97 | 376 |
Roughly, the set of $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
377 |
and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
|
378 |
\nn{need to also say something about collaring homeomorphisms.} |
|
379 |
\nn{this paragraph needs work.} |
|
380 |
||
103 | 381 |
Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
382 |
into a plain $n$-category (enriched over graded groups). |
|
97 | 383 |
\nn{say more here?} |
384 |
In the other direction, if we enrich over topological spaces instead of chain complexes, |
|
385 |
we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
|
386 |
instead of $C_*(\Homeo_\bd(X))$. |
|
387 |
Taking singular chains converts a space-type $A_\infty$ $n$-category into a chain complex |
|
388 |
type $A_\infty$ $n$-category. |
|
389 |
||
99 | 390 |
\medskip |
97 | 391 |
|
99 | 392 |
The alert reader will have already noticed that our definition of (plain) $n$-category |
393 |
is extremely similar to our definition of topological fields. |
|
142 | 394 |
The main difference is that for the $n$-category definition we restrict our attention to balls |
99 | 395 |
(and their boundaries), while for fields we consider all manifolds. |
142 | 396 |
(A minor difference is that in the category definition we directly impose isotopy |
397 |
invariance in dimension $n$, while in the fields definition we have non-isotopy-invariant fields |
|
398 |
but then mod out by local relations which imply isotopy invariance.) |
|
99 | 399 |
Thus a system of fields determines an $n$-category simply by restricting our attention to |
400 |
balls. |
|
142 | 401 |
This $n$-category can be thought of as the local part of the fields. |
99 | 402 |
Conversely, given an $n$-category we can construct a system of fields via |
142 | 403 |
a colimit construction; see below. |
99 | 404 |
|
142 | 405 |
%\nn{Next, say something about $A_\infty$ $n$-categories and ``homological" systems |
406 |
%of fields. |
|
407 |
%The universal (colimit) construction becomes our generalized definition of blob homology. |
|
408 |
%Need to explain how it relates to the old definition.} |
|
97 | 409 |
|
95 | 410 |
\medskip |
411 |
||
101 | 412 |
\nn{these examples need to be fleshed out a bit more} |
413 |
||
414 |
Examples of plain $n$-categories: |
|
415 |
\begin{itemize} |
|
416 |
||
417 |
\item Let $F$ be a closed $m$-manifold (e.g.\ a point). |
|
418 |
Let $T$ be a topological space. |
|
419 |
For $X$ a $k$-ball or $k$-sphere with $k < n$, define $\cC(X)$ to be the set of |
|
420 |
all maps from $X\times F$ to $T$. |
|
421 |
For $X$ an $n$-ball define $\cC(X)$ to be maps from $X\times F$ to $T$ modulo |
|
103 | 422 |
homotopies fixed on $\bd X \times F$. |
101 | 423 |
(Note that homotopy invariance implies isotopy invariance.) |
424 |
For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
|
425 |
be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. |
|
426 |
||
427 |
\item We can linearize the above example as follows. |
|
428 |
Let $\alpha$ be an $(n{+}m{+}1)$-cocycle on $T$ with values in a ring $R$ |
|
429 |
(e.g.\ the trivial cocycle). |
|
430 |
For $X$ of dimension less than $n$ define $\cC(X)$ as before. |
|
431 |
For $X$ an $n$-ball and $c\in \cC(\bd X)$ define $\cC(X; c)$ to be |
|
432 |
the $R$-module of finite linear combinations of maps from $X\times F$ to $T$, |
|
433 |
modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy |
|
434 |
$h: X\times F\times I \to T$, then $a \sim \alpha(h)b$. |
|
435 |
\nn{need to say something about fundamental classes, or choose $\alpha$ carefully} |
|
436 |
||
437 |
\item Given a traditional $n$-category $C$ (with strong duality etc.), |
|
438 |
define $\cC(X)$ (with $\dim(X) < n$) |
|
439 |
to be the set of all $C$-labeled sub cell complexes of $X$. |
|
142 | 440 |
(See Subsection \ref{sec:fields}.) |
101 | 441 |
For $X$ an $n$-ball and $c\in \cC(\bd X)$, define $\cC(X)$ to finite linear |
442 |
combinations of $C$-labeled sub cell complexes of $X$ |
|
443 |
modulo the kernel of the evaluation map. |
|
444 |
Define a product morphism $a\times D$ to be the product of the cell complex of $a$ with $D$, |
|
445 |
and with the same labeling as $a$. |
|
102 | 446 |
More generally, start with an $n{+}m$-category $C$ and a closed $m$-manifold $F$. |
447 |
Define $\cC(X)$, for $\dim(X) < n$, |
|
448 |
to be the set of all $C$-labeled sub cell complexes of $X\times F$. |
|
449 |
Define $\cC(X; c)$, for $X$ an $n$-ball, |
|
450 |
to be the dual Hilbert space $A(X\times F; c)$. |
|
101 | 451 |
\nn{refer elsewhere for details?} |
452 |
||
453 |
\item Variation on the above examples: |
|
103 | 454 |
We could allow $F$ to have boundary and specify boundary conditions on $X\times \bd F$, |
101 | 455 |
for example product boundary conditions or take the union over all boundary conditions. |
142 | 456 |
%\nn{maybe should not emphasize this case, since it's ``better" in some sense |
457 |
%to think of these guys as affording a representation |
|
458 |
%of the $n{+}1$-category associated to $\bd F$.} |
|
101 | 459 |
|
142 | 460 |
\item Here's our version of the bordism $n$-category. |
461 |
For a $k$-ball $X$, $k<n$, define $\cC(X)$ to be the set of all $k$-dimensional |
|
462 |
submanifolds $W$ of $X\times \r^\infty$ such that the projection $W \to X$ is transverse |
|
463 |
to $\bd X$. |
|
464 |
For $k=n$ define $\cC(X)$ to be homeomorphism classes (rel boundary) of such submanifolds; |
|
465 |
we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism |
|
466 |
$W\to W'$ which restricts to the identity on the boundary. |
|
467 |
||
143 | 468 |
\item \nn{sphere modules; ref to below} |
125 | 469 |
|
101 | 470 |
\end{itemize} |
471 |
||
472 |
||
473 |
Examples of $A_\infty$ $n$-categories: |
|
474 |
\begin{itemize} |
|
475 |
||
476 |
\item Same as in example \nn{xxxx} above (fiber $F$, target space $T$), |
|
477 |
but we define, for an $n$-ball $X$, $\cC(X; c)$ to be the chain complex |
|
478 |
$C_*(\Maps_c(X\times F))$, where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
|
479 |
and $C_*$ denotes singular chains. |
|
480 |
||
481 |
\item |
|
482 |
Given a plain $n$-category $C$, |
|
483 |
define $\cC(X; c) = \bc^C_*(X\times F; c)$, where $X$ is an $n$-ball |
|
484 |
and $\bc^C_*$ denotes the blob complex based on $C$. |
|
485 |
||
125 | 486 |
\item \nn{should add $\infty$ version of bordism $n$-cat} |
487 |
||
101 | 488 |
\end{itemize} |
95 | 489 |
|
108 | 490 |
|
491 |
||
492 |
||
493 |
||
494 |
||
495 |
\subsection{From $n$-categories to systems of fields} |
|
113 | 496 |
\label{ss:ncat_fields} |
108 | 497 |
|
498 |
We can extend the functors $\cC$ above from $k$-balls to arbitrary $k$-manifolds as follows. |
|
499 |
||
500 |
Let $W$ be a $k$-manifold, $1\le k \le n$. |
|
501 |
We will define a set $\cC(W)$. |
|
502 |
(If $k = n$ and our $k$-categories are enriched, then |
|
503 |
$\cC(W)$ will have additional structure; see below.) |
|
504 |
$\cC(W)$ will be the colimit of a functor defined on a category $\cJ(W)$, |
|
505 |
which we define next. |
|
506 |
||
142 | 507 |
Define a permissible decomposition of $W$ to be a cell decomposition |
108 | 508 |
\[ |
509 |
W = \bigcup_a X_a , |
|
510 |
\] |
|
142 | 511 |
where each closed top-dimensional cell $X_a$ is an embedded $k$-ball. |
108 | 512 |
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
513 |
of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
|
514 |
This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
|
515 |
(The objects of $\cJ(W)$ are permissible decompositions of $W$, and there is a unique |
|
119 | 516 |
morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
517 |
See Figure \ref{partofJfig}.) |
|
518 |
||
519 |
\begin{figure}[!ht] |
|
520 |
\begin{equation*} |
|
521 |
\mathfig{.63}{tempkw/zz2} |
|
522 |
\end{equation*} |
|
523 |
\caption{A small part of $\cJ(W)$} |
|
524 |
\label{partofJfig} |
|
525 |
\end{figure} |
|
526 |
||
108 | 527 |
|
528 |
$\cC$ determines |
|
529 |
a functor $\psi_\cC$ from $\cJ(W)$ to the category of sets |
|
530 |
(possibly with additional structure if $k=n$). |
|
531 |
For a decomposition $x = (X_a)$ in $\cJ(W)$, define $\psi_\cC(x)$ to be the subset |
|
532 |
\[ |
|
533 |
\psi_\cC(x) \sub \prod_a \cC(X_a) |
|
534 |
\] |
|
535 |
such that the restrictions to the various pieces of shared boundaries amongst the |
|
536 |
$X_a$ all agree. |
|
537 |
(Think fibered product.) |
|
538 |
If $x$ is a refinement of $y$, define a map $\psi_\cC(x)\to\psi_\cC(y)$ |
|
539 |
via the composition maps of $\cC$. |
|
112 | 540 |
(If $\dim(W) = n$ then we need to also make use of the monoidal |
541 |
product in the enriching category. |
|
542 |
\nn{should probably be more explicit here}) |
|
108 | 543 |
|
544 |
Finally, define $\cC(W)$ to be the colimit of $\psi_\cC$. |
|
142 | 545 |
When $k<n$ or $k=n$ and we are in the plain (non-$A_\infty$) case, this means that |
112 | 546 |
for each decomposition $x$ there is a map |
108 | 547 |
$\psi_\cC(x)\to \cC(W)$, these maps are compatible with the refinement maps |
548 |
above, and $\cC(W)$ is universal with respect to these properties. |
|
142 | 549 |
When $k=n$ and we are in the $A_\infty$ case, it means |
550 |
homotopy colimit. |
|
112 | 551 |
|
552 |
More concretely, in the plain case enriched over vector spaces, and with $\dim(W) = n$, we can take |
|
553 |
\[ |
|
554 |
\cC(W) = \left( \oplus_x \psi_\cC(x)\right) \big/ K |
|
555 |
\] |
|
556 |
where $K$ is generated by all things of the form $a - g(a)$, where |
|
557 |
$a\in \psi_\cC(x)$ for some decomposition $x$, $x\le y$, and $g: \psi_\cC(x) |
|
558 |
\to \psi_\cC(y)$ is value of $\psi_\cC$ on the antirefinement $x\to y$. |
|
111 | 559 |
|
112 | 560 |
In the $A_\infty$ case enriched over chain complexes, the concrete description of the colimit |
561 |
is as follows. |
|
142 | 562 |
%\nn{should probably rewrite this to be compatible with some standard reference} |
113 | 563 |
Define an $m$-sequence to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions. |
112 | 564 |
Such sequences (for all $m$) form a simplicial set. |
565 |
Let |
|
566 |
\[ |
|
567 |
V = \bigoplus_{(x_i)} \psi_\cC(x_0) , |
|
568 |
\] |
|
113 | 569 |
where the sum is over all $m$-sequences and all $m$, and each summand is degree shifted by $m$. |
112 | 570 |
We endow $V$ with a differential which is the sum of the differential of the $\psi_\cC(x_0)$ |
571 |
summands plus another term using the differential of the simplicial set of $m$-sequences. |
|
572 |
More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
|
573 |
summand of $V$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
|
574 |
\[ |
|
575 |
\bd (a, \bar{x}) = (\bd a, \bar{x}) \pm (g(a), d_0(\bar{x})) + \sum_{j=1}^k \pm (a, d_j(\bar{x})) , |
|
576 |
\] |
|
577 |
where $d_j(\bar{x}) = (x_0,\dots,x_{j-1},x_{j+1},\dots,x_k)$ and $g: \psi_\cC(x_0)\to \psi_\cC(x_1)$ |
|
578 |
is the usual map. |
|
579 |
\nn{need to say this better} |
|
580 |
\nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which |
|
581 |
combine only two balls at a time; for $n=1$ this version will lead to usual definition |
|
582 |
of $A_\infty$ category} |
|
108 | 583 |
|
113 | 584 |
We will call $m$ the filtration degree of the complex. |
585 |
We can think of this construction as starting with a disjoint copy of a complex for each |
|
586 |
permissible decomposition (filtration degree 0). |
|
587 |
Then we glue these together with mapping cylinders coming from gluing maps |
|
588 |
(filtration degree 1). |
|
589 |
Then we kill the extra homology we just introduced with mapping cylinder between the mapping cylinders (filtration degree 2). |
|
590 |
And so on. |
|
591 |
||
108 | 592 |
$\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
593 |
||
594 |
It is easy to see that |
|
595 |
there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps |
|
596 |
comprise a natural transformation of functors. |
|
597 |
||
598 |
\nn{need to finish explaining why we have a system of fields; |
|
599 |
need to say more about ``homological" fields? |
|
600 |
(actions of homeomorphisms); |
|
601 |
define $k$-cat $\cC(\cdot\times W)$} |
|
602 |
||
603 |
||
604 |
||
605 |
\subsection{Modules} |
|
95 | 606 |
|
101 | 607 |
Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations, |
608 |
a.k.a.\ actions). |
|
102 | 609 |
The definition will be very similar to that of $n$-categories. |
109 | 610 |
\nn{** need to make sure all revisions of $n$-cat def are also made to module def.} |
110 | 611 |
\nn{should they be called $n$-modules instead of just modules? probably not, but worth considering.} |
102 | 612 |
|
104 | 613 |
Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
102 | 614 |
in the context of an $m{+}1$-dimensional TQFT. |
615 |
Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
|
616 |
This will be explained in more detail as we present the axioms. |
|
617 |
||
618 |
Fix an $n$-category $\cC$. |
|
619 |
||
620 |
Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair |
|
621 |
(standard $k$-ball, northern hemisphere in boundary of standard $k$-ball). |
|
622 |
We call $B$ the ball and $N$ the marking. |
|
623 |
A homeomorphism between marked $k$-balls is a homeomorphism of balls which |
|
624 |
restricts to a homeomorphism of markings. |
|
625 |
||
626 |
\xxpar{Module morphisms} |
|
627 |
{For each $0 \le k \le n$, we have a functor $\cM_k$ from |
|
628 |
the category of marked $k$-balls and |
|
629 |
homeomorphisms to the category of sets and bijections.} |
|
630 |
||
631 |
(As with $n$-categories, we will usually omit the subscript $k$.) |
|
632 |
||
104 | 633 |
For example, let $\cD$ be the $m{+}1$-dimensional TQFT which assigns to a $k$-manifold $N$ the set |
634 |
of maps from $N$ to $T$, modulo homotopy (and possibly linearized) if $k=m$. |
|
635 |
Let $W$ be an $(m{-}n{+}1)$-dimensional manifold with boundary. |
|
636 |
Let $\cC$ be the $n$-category with $\cC(X) \deq \cD(X\times \bd W)$. |
|
637 |
Let $\cM(B, N) \deq \cD((B\times \bd W)\cup (N\times W))$. |
|
638 |
(The union is along $N\times \bd W$.) |
|
110 | 639 |
(If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be |
640 |
the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.) |
|
102 | 641 |
|
182 | 642 |
\begin{figure}[!ht] |
643 |
$$\mathfig{.8}{tempkw/blah15}$$ |
|
644 |
\caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure} |
|
645 |
||
103 | 646 |
Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
647 |
Call such a thing a {marked $k{-}1$-hemisphere}. |
|
102 | 648 |
|
649 |
\xxpar{Module boundaries, part 1:} |
|
650 |
{For each $0 \le k \le n-1$, we have a functor $\cM_k$ from |
|
104 | 651 |
the category of marked $k$-hemispheres and |
102 | 652 |
homeomorphisms to the category of sets and bijections.} |
653 |
||
104 | 654 |
In our example, let $\cM(H) \deq \cD(H\times\bd W \cup \bd H\times W)$. |
655 |
||
102 | 656 |
\xxpar{Module boundaries, part 2:} |
657 |
{For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cM(\bd M)$. |
|
658 |
These maps, for various $M$, comprise a natural transformation of functors.} |
|
659 |
||
110 | 660 |
Given $c\in\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
102 | 661 |
|
662 |
If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
|
663 |
then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$ |
|
664 |
and $c\in \cC(\bd M)$. |
|
665 |
||
666 |
\xxpar{Module domain $+$ range $\to$ boundary:} |
|
667 |
{Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k$-hemisphere ($0\le k\le n-1$), |
|
104 | 668 |
$M_i$ is a marked $k$-ball, and $E = M_1\cap M_2$ is a marked $k{-}1$-hemisphere. |
669 |
Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the |
|
670 |
two maps $\bd: \cM(M_i)\to \cM(E)$. |
|
102 | 671 |
Then (axiom) we have an injective map |
672 |
\[ |
|
673 |
\gl_E : \cM(M_1) \times_{\cM(E)} \cM(M_2) \to \cM(H) |
|
674 |
\] |
|
675 |
which is natural with respect to the actions of homeomorphisms.} |
|
676 |
||
110 | 677 |
Let $\cM(H)_E$ denote the image of $\gl_E$. |
678 |
We will refer to elements of $\cM(H)_E$ as ``splittable along $E$" or ``transverse to $E$". |
|
679 |
||
680 |
||
103 | 681 |
\xxpar{Axiom yet to be named:} |
682 |
{For each marked $k$-hemisphere $H$ there is a restriction map |
|
683 |
$\cM(H)\to \cC(H)$. |
|
684 |
($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) |
|
685 |
These maps comprise a natural transformation of functors.} |
|
102 | 686 |
|
103 | 687 |
Note that combining the various boundary and restriction maps above |
110 | 688 |
(for both modules and $n$-categories) |
103 | 689 |
we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$ |
690 |
a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$. |
|
110 | 691 |
The subset is the subset of morphisms which are appropriately splittable (transverse to the |
692 |
cutting submanifolds). |
|
103 | 693 |
This fact will be used below. |
102 | 694 |
|
104 | 695 |
In our example, the various restriction and gluing maps above come from |
696 |
restricting and gluing maps into $T$. |
|
697 |
||
698 |
We require two sorts of composition (gluing) for modules, corresponding to two ways |
|
103 | 699 |
of splitting a marked $k$-ball into two (marked or plain) $k$-balls. |
119 | 700 |
(See Figure \ref{zzz3}.) |
103 | 701 |
|
119 | 702 |
\begin{figure}[!ht] |
703 |
\begin{equation*} |
|
704 |
\mathfig{.63}{tempkw/zz3} |
|
705 |
\end{equation*} |
|
706 |
\caption{Module composition (top); $n$-category action (bottom)} |
|
707 |
\label{zzz3} |
|
708 |
\end{figure} |
|
709 |
||
710 |
First, we can compose two module morphisms to get another module morphism. |
|
103 | 711 |
|
712 |
\xxpar{Module composition:} |
|
713 |
{Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls ($0\le k\le n$) |
|
714 |
and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball. |
|
715 |
Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere. |
|
716 |
Note that each of $M$, $M_1$ and $M_2$ has its boundary split into two marked $k{-}1$-balls by $E$. |
|
717 |
We have restriction (domain or range) maps $\cM(M_i)_E \to \cM(Y)$. |
|
718 |
Let $\cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E$ denote the fibered product of these two maps. |
|
719 |
Then (axiom) we have a map |
|
720 |
\[ |
|
721 |
\gl_Y : \cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E \to \cM(M)_E |
|
722 |
\] |
|
723 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
724 |
to the intersection of the boundaries of $M$ and $M_i$. |
|
725 |
If $k < n$ we require that $\gl_Y$ is injective. |
|
726 |
(For $k=n$, see below.)} |
|
727 |
||
119 | 728 |
|
729 |
||
103 | 730 |
Second, we can compose an $n$-category morphism with a module morphism to get another |
731 |
module morphism. |
|
732 |
We'll call this the action map to distinguish it from the other kind of composition. |
|
733 |
||
734 |
\xxpar{$n$-category action:} |
|
735 |
{Let $M = X \cup_Y M'$, where $M$ and $M'$ are marked $k$-balls ($0\le k\le n$), |
|
736 |
$X$ is a plain $k$-ball, |
|
737 |
and $Y = X\cap M'$ is a $k{-}1$-ball. |
|
738 |
Let $E = \bd Y$, which is a $k{-}2$-sphere. |
|
739 |
We have restriction maps $\cM(M')_E \to \cC(Y)$ and $\cC(X)_E\to \cC(Y)$. |
|
740 |
Let $\cC(X)_E \times_{\cC(Y)} \cM(M')_E$ denote the fibered product of these two maps. |
|
741 |
Then (axiom) we have a map |
|
742 |
\[ |
|
743 |
\gl_Y :\cC(X)_E \times_{\cC(Y)} \cM(M')_E \to \cM(M)_E |
|
744 |
\] |
|
745 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
746 |
to the intersection of the boundaries of $X$ and $M'$. |
|
747 |
If $k < n$ we require that $\gl_Y$ is injective. |
|
748 |
(For $k=n$, see below.)} |
|
749 |
||
750 |
\xxpar{Module strict associativity:} |
|
751 |
{The composition and action maps above are strictly associative.} |
|
752 |
||
110 | 753 |
Note that the above associativity axiom applies to mixtures of module composition, |
754 |
action maps and $n$-category composition. |
|
119 | 755 |
See Figure \ref{zzz1b}. |
756 |
||
757 |
\begin{figure}[!ht] |
|
758 |
\begin{equation*} |
|
759 |
\mathfig{1}{tempkw/zz1b} |
|
760 |
\end{equation*} |
|
761 |
\caption{Two examples of mixed associativity} |
|
762 |
\label{zzz1b} |
|
763 |
\end{figure} |
|
764 |
||
110 | 765 |
|
766 |
The above three axioms are equivalent to the following axiom, |
|
103 | 767 |
which we state in slightly vague form. |
768 |
\nn{need figure for this} |
|
769 |
||
770 |
\xxpar{Module multi-composition:} |
|
771 |
{Given any decomposition |
|
772 |
\[ |
|
773 |
M = X_1 \cup\cdots\cup X_p \cup M_1\cup\cdots\cup M_q |
|
774 |
\] |
|
775 |
of a marked $k$-ball $M$ |
|
776 |
into small (marked and plain) $k$-balls $M_i$ and $X_j$, there is a |
|
777 |
map from an appropriate subset (like a fibered product) |
|
778 |
of |
|
779 |
\[ |
|
780 |
\cC(X_1)\times\cdots\times\cC(X_p) \times \cM(M_1)\times\cdots\times\cM(M_q) |
|
781 |
\] |
|
782 |
to $\cM(M)$, |
|
783 |
and these various multifold composition maps satisfy an |
|
784 |
operad-type strict associativity condition.} |
|
785 |
||
786 |
(The above operad-like structure is analogous to the swiss cheese operad |
|
146 | 787 |
\cite{MR1718089}.) |
103 | 788 |
\nn{need to double-check that this is true.} |
789 |
||
790 |
\xxpar{Module product (identity) morphisms:} |
|
791 |
{Let $M$ be a marked $k$-ball and $D$ be a plain $m$-ball, with $k+m \le n$. |
|
792 |
Then we have a map $\cM(M)\to \cM(M\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cM(M)$. |
|
793 |
If $f:M\to M'$ and $\tilde{f}:M\times D \to M'\times D'$ are maps such that the diagram |
|
794 |
\[ \xymatrix{ |
|
795 |
M\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & M'\times D' \ar[d]^{\pi} \\ |
|
796 |
M \ar[r]^{f} & M' |
|
797 |
} \] |
|
798 |
commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.} |
|
799 |
||
111 | 800 |
\nn{Need to add compatibility with various things, as in the n-cat version of this axiom above.} |
103 | 801 |
|
110 | 802 |
\nn{** marker --- resume revising here **} |
803 |
||
103 | 804 |
There are two alternatives for the next axiom, according whether we are defining |
805 |
modules for plain $n$-categories or $A_\infty$ $n$-categories. |
|
806 |
In the plain case we require |
|
807 |
||
185 | 808 |
\xxpar{Extended isotopy invariance in dimension $n$:} |
103 | 809 |
{Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts |
175 | 810 |
to the identity on $\bd M$ and is extended isotopic (rel boundary) to the identity. |
103 | 811 |
Then $f$ acts trivially on $\cM(M)$.} |
812 |
||
813 |
\nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
|
814 |
||
815 |
We emphasize that the $\bd M$ above means boundary in the marked $k$-ball sense. |
|
816 |
In other words, if $M = (B, N)$ then we require only that isotopies are fixed |
|
817 |
on $\bd B \setmin N$. |
|
818 |
||
819 |
For $A_\infty$ modules we require |
|
820 |
||
821 |
\xxpar{Families of homeomorphisms act.} |
|
822 |
{For each marked $n$-ball $M$ and each $c\in \cM(\bd M)$ we have a map of chain complexes |
|
823 |
\[ |
|
824 |
C_*(\Homeo_\bd(M))\ot \cM(M; c) \to \cM(M; c) . |
|
825 |
\] |
|
826 |
Here $C_*$ means singular chains and $\Homeo_\bd(M)$ is the space of homeomorphisms of $M$ |
|
827 |
which fix $\bd M$. |
|
828 |
These action maps are required to be associative up to homotopy |
|
829 |
\nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
|
830 |
a diagram like the one in Proposition \ref{CDprop} commutes. |
|
831 |
\nn{repeat diagram here?} |
|
832 |
\nn{restate this with $\Homeo(M\to M')$? what about boundary fixing property?}} |
|
833 |
||
834 |
\medskip |
|
102 | 835 |
|
104 | 836 |
Note that the above axioms imply that an $n$-category module has the structure |
837 |
of an $n{-}1$-category. |
|
838 |
More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$, |
|
839 |
where $X$ is a $k$-ball or $k{-}1$-sphere and in the product $X\times J$ we pinch |
|
840 |
above the non-marked boundary component of $J$. |
|
841 |
\nn{give figure for this, or say more?} |
|
842 |
Then $\cE$ has the structure of an $n{-}1$-category. |
|
102 | 843 |
|
105 | 844 |
All marked $k$-balls are homeomorphic, unless $k = 1$ and our manifolds |
845 |
are oriented or Spin (but not unoriented or $\text{Pin}_\pm$). |
|
846 |
In this case ($k=1$ and oriented or Spin), there are two types |
|
847 |
of marked 1-balls, call them left-marked and right-marked, |
|
848 |
and hence there are two types of modules, call them right modules and left modules. |
|
849 |
In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$), |
|
850 |
there is no left/right module distinction. |
|
851 |
||
130 | 852 |
\medskip |
853 |
||
854 |
Examples of modules: |
|
855 |
\begin{itemize} |
|
142 | 856 |
\item \nn{examples from TQFTs} |
857 |
\item \nn{for maps to $T$, can restrict to subspaces of $T$;} |
|
130 | 858 |
\end{itemize} |
108 | 859 |
|
860 |
\subsection{Modules as boundary labels} |
|
112 | 861 |
\label{moddecss} |
108 | 862 |
|
863 |
Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), |
|
143 | 864 |
let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$, |
865 |
and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$. |
|
866 |
||
867 |
%Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), |
|
868 |
%and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary |
|
869 |
%component $\bd_i W$ of $W$. |
|
870 |
%(More generally, each $\cN_i$ could label some codimension zero submanifold of $\bd W$.) |
|
108 | 871 |
|
872 |
We will define a set $\cC(W, \cN)$ using a colimit construction similar to above. |
|
873 |
\nn{give ref} |
|
874 |
(If $k = n$ and our $k$-categories are enriched, then |
|
875 |
$\cC(W, \cN)$ will have additional structure; see below.) |
|
876 |
||
877 |
Define a permissible decomposition of $W$ to be a decomposition |
|
878 |
\[ |
|
879 |
W = (\bigcup_a X_a) \cup (\bigcup_{i,b} M_{ib}) , |
|
880 |
\] |
|
881 |
where each $X_a$ is a plain $k$-ball (disjoint from $\bd W$) and |
|
882 |
each $M_{ib}$ is a marked $k$-ball intersecting $\bd_i W$, |
|
143 | 883 |
with $M_{ib}\cap Y_i$ being the marking. |
884 |
(See Figure \ref{mblabel}.) |
|
885 |
\begin{figure}[!ht]\begin{equation*} |
|
886 |
\mathfig{.9}{tempkw/mblabel} |
|
887 |
\end{equation*}\caption{A permissible decomposition of a manifold |
|
146 | 888 |
whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$.}\label{mblabel}\end{figure} |
108 | 889 |
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
890 |
of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
|
891 |
This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
|
892 |
(The objects of $\cJ(D)$ are permissible decompositions of $W$, and there is a unique |
|
893 |
morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
|
894 |
||
895 |
$\cN$ determines |
|
896 |
a functor $\psi_\cN$ from $\cJ(W)$ to the category of sets |
|
897 |
(possibly with additional structure if $k=n$). |
|
898 |
For a decomposition $x = (X_a, M_{ib})$ in $\cJ(W)$, define $\psi_\cN(x)$ to be the subset |
|
899 |
\[ |
|
111 | 900 |
\psi_\cN(x) \sub (\prod_a \cC(X_a)) \times (\prod_{ib} \cN_i(M_{ib})) |
108 | 901 |
\] |
902 |
such that the restrictions to the various pieces of shared boundaries amongst the |
|
903 |
$X_a$ and $M_{ib}$ all agree. |
|
904 |
(Think fibered product.) |
|
905 |
If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$ |
|
906 |
via the gluing (composition or action) maps from $\cC$ and the $\cN_i$. |
|
907 |
||
908 |
Finally, define $\cC(W, \cN)$ to be the colimit of $\psi_\cN$. |
|
143 | 909 |
(Recall that if $k=n$ and we are in the $A_\infty$ case, then ``colimit" means |
910 |
homotopy colimit.) |
|
108 | 911 |
|
143 | 912 |
If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define |
913 |
$\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold |
|
914 |
$D\times Y_i \sub \bd(D\times W)$. |
|
112 | 915 |
|
143 | 916 |
It is not hard to see that the assignment $D \mapsto \cT(W, \cN)(D) \deq \cC(D\times W, \cN)$ |
917 |
has the structure of an $n{-}k$-category. |
|
144 | 918 |
|
919 |
\medskip |
|
920 |
||
921 |
||
922 |
%\subsection{Tensor products} |
|
108 | 923 |
|
144 | 924 |
We will use a simple special case of the above |
925 |
construction to define tensor products |
|
926 |
of modules. |
|
927 |
Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$. |
|
928 |
(If $k=1$ and manifolds are oriented, then one should be |
|
929 |
a left module and the other a right module.) |
|
930 |
Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$. |
|
931 |
Define the tensor product of $\cM_1$ and $\cM_2$ to be the |
|
932 |
$n{-}1$-category $\cT(J, \cM_1, \cM_2)$, |
|
933 |
\[ |
|
934 |
\cM_1\otimes \cM_2 \deq \cT(J, \cM_1, \cM_2) . |
|
935 |
\] |
|
936 |
This of course depends (functorially) |
|
937 |
on the choice of 1-ball $J$. |
|
105 | 938 |
|
144 | 939 |
We will define a more general self tensor product (categorified coend) below. |
940 |
||
112 | 941 |
|
144 | 942 |
|
943 |
||
944 |
%\nn{what about self tensor products /coends ?} |
|
105 | 945 |
|
108 | 946 |
\nn{maybe ``tensor product" is not the best name?} |
947 |
||
144 | 948 |
%\nn{start with (less general) tensor products; maybe change this later} |
106 | 949 |
|
107 | 950 |
|
951 |
||
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|
952 |
\subsection{The $n{+}1$-category of sphere modules} |
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|
953 |
|
155 | 954 |
In this subsection we define an $n{+}1$-category of ``sphere modules" whose objects |
955 |
correspond to $n$-categories. |
|
956 |
This is a version of the familiar algebras-bimodules-intertwinors 2-category. |
|
957 |
(Terminology: It is clearly appropriate to call an $S^0$ modules a bimodule, |
|
958 |
since a 0-sphere has an obvious bi-ness. |
|
959 |
This is much less true for higher dimensional spheres, |
|
960 |
so we prefer the term ``sphere module" for the general case.) |
|
144 | 961 |
|
107 | 962 |
|
963 |
||
144 | 964 |
\nn{need to assume a little extra structure to define the top ($n+1$) part (?)} |
101 | 965 |
|
966 |
\medskip |
|
967 |
\hrule |
|
968 |
\medskip |
|
969 |
||
95 | 970 |
\nn{to be continued...} |
101 | 971 |
\medskip |
98 | 972 |
|
973 |
||
974 |
Stuff that remains to be done (either below or in an appendix or in a separate section or in |
|
975 |
a separate paper): |
|
976 |
\begin{itemize} |
|
977 |
\item traditional $n$-cat defs (e.g. *-1-cat, pivotal 2-cat) imply our def of plain $n$-cat |
|
978 |
\item conversely, our def implies other defs |
|
105 | 979 |
\item do same for modules; maybe an appendix on relating topological |
980 |
vs traditional defs, $n = 1,2$, $A_\infty$ or not, cats, modules, tensor products |
|
98 | 981 |
\item traditional $A_\infty$ 1-cat def implies our def |
99 | 982 |
\item ... and vice-versa (already done in appendix) |
98 | 983 |
\item say something about unoriented vs oriented vs spin vs pin for $n=1$ (and $n=2$?) |
984 |
\item spell out what difference (if any) Top vs PL vs Smooth makes |
|
117
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|
985 |
\item define $n{+}1$-cat of $n$-cats (a.k.a.\ $n{+}1$-category of generalized bimodules |
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|
986 |
a.k.a.\ $n{+}1$-category of sphere modules); discuss Morita equivalence |
130 | 987 |
\item morphisms of modules; show that it's adjoint to tensor product |
139 | 988 |
(need to define dual module for this) |
989 |
\item functors |
|
98 | 990 |
\end{itemize} |
991 |
||
134 | 992 |
\nn{Some salvaged paragraphs that we might want to work back in:} |
993 |
\hrule |
|
98 | 994 |
|
134 | 995 |
Appendix \ref{sec:comparing-A-infty} explains the translation between this definition of an $A_\infty$ $1$-category and the usual one expressed in terms of `associativity up to higher homotopy', as in \cite{MR1854636}. (In this version of the paper, that appendix is incomplete, however.) |
996 |
||
997 |
The motivating example is `chains of maps to $M$' for some fixed target space $M$. This is a topological $A_\infty$ category $\Xi_M$ with $\Xi_M(J) = C_*(\Maps(J \to M))$. The gluing maps $\Xi_M(J) \tensor \Xi_M(J') \to \Xi_M(J \cup J')$ takes the product of singular chains, then glues maps to $M$ together; the associativity condition is automatically satisfied. The evaluation map $\ev_{J,J'} : \CD{J \to J'} \tensor \Xi_M(J) \to \Xi_M(J')$ is the composition |
|
998 |
\begin{align*} |
|
999 |
\CD{J \to J'} \tensor C_*(\Maps(J \to M)) & \to C_*(\Diff(J \to J') \times \Maps(J \to M)) \\ & \to C_*(\Maps(J' \to M)), |
|
1000 |
\end{align*} |
|
1001 |
where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism. |
|
1002 |
||
1003 |
We now give two motivating examples, as theorems constructing other homological systems of fields, |
|
1004 |
||
1005 |
||
1006 |
\begin{thm} |
|
1007 |
For a fixed target space $X$, `chains of maps to $X$' is a homological system of fields $\Xi$, defined as |
|
1008 |
\begin{equation*} |
|
1009 |
\Xi(M) = \CM{M}{X}. |
|
1010 |
\end{equation*} |
|
1011 |
\end{thm} |
|
1012 |
||
1013 |
\begin{thm} |
|
1014 |
Given an $n$-dimensional system of fields $\cF$, and a $k$-manifold $F$, there is an $n-k$ dimensional homological system of fields $\cF^{\times F}$ defined by |
|
1015 |
\begin{equation*} |
|
1016 |
\cF^{\times F}(M) = \cB_*(M \times F, \cF). |
|
1017 |
\end{equation*} |
|
1018 |
\end{thm} |
|
1019 |
We might suggestively write $\cF^{\times F}$ as $\cB_*(F \times [0,1]^b, \cF)$, interpreting this as an (undefined!) $A_\infty$ $b$-category, and then as the resulting homological system of fields, following a recipe analogous to that given above for $A_\infty$ $1$-categories. |
|
1020 |
||
1021 |
||
1022 |
In later sections, we'll prove the following two unsurprising theorems, about the (as-yet-undefined) blob homology of these homological systems of fields. |
|
1023 |
||
1024 |
||
1025 |
\begin{thm} |
|
1026 |
\begin{equation*} |
|
1027 |
\cB_*(M, \Xi) \iso \Xi(M) |
|
1028 |
\end{equation*} |
|
1029 |
\end{thm} |
|
1030 |
||
1031 |
\begin{thm}[Product formula] |
|
1032 |
Given a $b$-manifold $B$, an $f$-manifold $F$ and a $b+f$ dimensional system of fields, |
|
1033 |
there is a quasi-isomorphism |
|
1034 |
\begin{align*} |
|
1035 |
\cB_*(B \times F, \cF) & \quismto \cB_*(B, \cF^{\times F}) |
|
1036 |
\end{align*} |
|
1037 |
\end{thm} |
|
1038 |
||
1039 |
\begin{question} |
|
1040 |
Is it possible to compute the blob homology of a non-trivial bundle in terms of the blob homology of its fiber? |
|
1041 |
\end{question} |
|
1042 |
||
1043 |
\hrule |