author | Kevin Walker <kevin@canyon23.net> |
Tue, 01 Jun 2010 11:08:17 -0700 | |
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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{$n$-categories and their modules} |
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\label{sec:ncats} |
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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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There are many existing definitions of $n$-categories, with various intended uses. In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. Generally, these sets are indexed by instances of a certain typical chape. |
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Some $n$-category definitions model $k$-morphisms on the standard bihedrons (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.) |
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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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\begin{axiom}[Morphisms]{\textup{\textbf{[preliminary]}}} |
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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{axiom} |
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By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
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standard $k$-ball. |
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We {\it do not} assume that it is equipped with a |
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preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below. |
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Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on |
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the boundary), we want a corresponding |
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bijection of sets $f:\cC(X)\to \cC(Y)$. |
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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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\addtocounter{axiom}{-1} |
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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 $k$-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.) |
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For each flavor of manifold there is a corresponding flavor of $n$-category. |
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We will concentrate on the case of PL unoriented manifolds. |
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(The 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 $Y$. |
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This will be used below 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 the tangent bundle, or the frame |
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bundle (i.e.\ framed balls), or more generally some 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-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 division into domain and range makes less sense. For example, in a pivotal tensor category, there are natural isomorphisms $\Hom{}{A}{B \tensor C} \isoto \Hom{}{B^* \tensor A}{C}$, etc. (sometimes called ``Frobenius reciprocity''), which canonically identify all the morphism spaces which have the same boundary. We prefer to not 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 the subsection for $0$ through $k-1$ we can use a coend |
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construction, as described in Subsection \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{prop} |
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\label{axiom:spheres} |
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For each $1 \le k \le n$, we have a functor $\cC_{k-1}$ 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{prop} |
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We postpone the proof of this result until after we've actually given all the axioms. Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, along with the data described in 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 $\bd: \cC_k(X)\to \cC_{k-1}(\bd X)$. |
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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 $c\in\cC(\bd(X))$, 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 $c\in \cC(\bd X)$, 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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The following proposition follows from the coend construction used to define $\cC_{k-1}$ |
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on spheres. |
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\begin{prop}[Boundary from 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}). |
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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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\[ |
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\gl_E : \cC(B_1) \times_{\cC(E)} \cC(B_2) \into \cC(S) |
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\] |
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which is natural with respect to the actions of homeomorphisms. |
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(When $k=1$ we stipulate that $\cC(E)$ is a point, so that the above fibered product |
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becomes a normal product.) |
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\end{prop} |
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\begin{figure}[!ht] |
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$$ |
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\begin{tikzpicture}[%every label/.style={green} |
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] |
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\node[fill=black, circle, label=below:$E$, inner sep=2pt](S) at (0,0) {}; |
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\node[fill=black, circle, label=above:$E$, inner sep=2pt](N) at (0,2) {}; |
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\draw (S) arc (-90:90:1); |
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\draw (N) arc (90:270:1); |
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\node[left] at (-1,1) {$B_1$}; |
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\node[right] at (1,1) {$B_2$}; |
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\end{tikzpicture} |
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$$ |
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\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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If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ |
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as above, then we define $\cC(X)_E = \bd^{-1}(\cC(\bd X)_E)$. |
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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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More generally, we also include under the rubric ``restriction map" the |
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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. |
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In particular, we have restriction maps $\cC(X)_E \to \cC(B_i)$ |
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($i = 1, 2$, notation from previous paragraph). |
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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 $\bd X = B_1 \cup_E B_2$. |
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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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We have a map |
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\[ |
223 |
\gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E |
|
224 |
\] |
|
225 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
226 |
to the intersection of the boundaries of $B$ and $B_i$. |
|
227 |
If $k < n$ we require that $\gl_Y$ is injective. |
|
187 | 228 |
(For $k=n$, see below.) |
229 |
\end{axiom} |
|
94 | 230 |
|
179 | 231 |
\begin{figure}[!ht] |
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$$ |
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\begin{tikzpicture}[%every label/.style={green}, |
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x=1.5cm,y=1.5cm] |
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\node[fill=black, circle, label=below:$E$, inner sep=2pt](S) at (0,0) {}; |
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\node[fill=black, circle, label=above:$E$, inner sep=2pt](N) at (0,2) {}; |
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\draw (S) arc (-90:90:1); |
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\draw (N) arc (90:270:1); |
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\draw (N) -- (S); |
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\node[left] at (-1/4,1) {$B_1$}; |
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\node[right] at (1/4,1) {$B_2$}; |
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\node at (1/6,3/2) {$Y$}; |
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\end{tikzpicture} |
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$$ |
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\caption{From two balls to one ball.}\label{blah5}\end{figure} |
179 | 246 |
|
195 | 247 |
\begin{axiom}[Strict associativity] \label{nca-assoc} |
187 | 248 |
The composition (gluing) maps above are strictly associative. |
249 |
\end{axiom} |
|
102 | 250 |
|
179 | 251 |
\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 | 254 |
|
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We'll use the notations $a\bullet b$ as well as $a \cup b$ for the glued together field $\gl_Y(a, b)$. |
110 | 256 |
In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ |
195 | 257 |
a restriction map (one of many types of map so called) and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$. |
258 |
%Compositions of boundary and restriction maps will also be called restriction maps. |
|
259 |
%For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a |
|
260 |
%restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$. |
|
110 | 261 |
|
192 | 262 |
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 `splittable along $Y$' or `transverse to $Y$'. |
192 | 264 |
We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$. |
109 | 265 |
|
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266 |
More generally, let $\alpha$ be a subdivision of a ball $X$ into smaller balls. |
193 | 267 |
Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from |
268 |
the smaller balls to $X$. |
|
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We say that elements of $\cC(X)_\alpha$ are morphisms which are `splittable along $\alpha$'. |
193 | 270 |
In situations where the subdivision is notationally anonymous, we will write |
271 |
$\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to) |
|
272 |
the unnamed subdivision. |
|
273 |
If $\beta$ is a subdivision of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cC(\bd X)_\beta)$; |
|
274 |
this can also be denoted $\cC(X)\spl$ if the context contains an anonymous |
|
275 |
subdivision of $\bd X$ and no competing subdivision of $X$. |
|
192 | 276 |
|
277 |
The above two composition axioms are equivalent to the following one, |
|
102 | 278 |
which we state in slightly vague form. |
279 |
||
280 |
\xxpar{Multi-composition:} |
|
281 |
{Given any decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball |
|
282 |
into small $k$-balls, there is a |
|
283 |
map from an appropriate subset (like a fibered product) |
|
193 | 284 |
of $\cC(B_1)\spl\times\cdots\times\cC(B_m)\spl$ to $\cC(B)\spl$, |
95 | 285 |
and these various $m$-fold composition maps satisfy an |
179 | 286 |
operad-type strict associativity condition (Figure \ref{blah7}).} |
287 |
||
288 |
\begin{figure}[!ht] |
|
289 |
$$\mathfig{.8}{tempkw/blah7}$$ |
|
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\caption{Operad composition and associativity}\label{blah7}\end{figure} |
95 | 291 |
|
292 |
The next axiom is related to identity morphisms, though that might not be immediately obvious. |
|
293 |
||
187 | 294 |
\begin{axiom}[Product (identity) morphisms] |
191
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For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. These maps must satisfy the following conditions. |
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\begin{enumerate} |
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\item |
95 | 298 |
If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram |
299 |
\[ \xymatrix{ |
|
96 | 300 |
X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\ |
95 | 301 |
X \ar[r]^{f} & X' |
302 |
} \] |
|
109 | 303 |
commutes, then we have |
304 |
\[ |
|
305 |
\tilde{f}(a\times D) = f(a)\times D' . |
|
306 |
\] |
|
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\item |
109 | 308 |
Product morphisms are compatible with gluing (composition) in both factors: |
309 |
\[ |
|
310 |
(a'\times D)\bullet(a''\times D) = (a'\bullet a'')\times D |
|
311 |
\] |
|
312 |
and |
|
313 |
\[ |
|
314 |
(a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') . |
|
315 |
\] |
|
122 | 316 |
\nn{if pinched boundary, then remove first case above} |
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\item |
109 | 318 |
Product morphisms are associative: |
319 |
\[ |
|
320 |
(a\times D)\times D' = a\times (D\times D') . |
|
321 |
\] |
|
322 |
(Here we are implicitly using functoriality and the obvious homeomorphism |
|
323 |
$(X\times D)\times D' \to X\times(D\times D')$.) |
|
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\item |
110 | 325 |
Product morphisms are compatible with restriction: |
326 |
\[ |
|
327 |
\res_{X\times E}(a\times D) = a\times E |
|
328 |
\] |
|
329 |
for $E\sub \bd D$ and $a\in \cC(X)$. |
|
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\end{enumerate} |
187 | 331 |
\end{axiom} |
95 | 332 |
|
110 | 333 |
\nn{need even more subaxioms for product morphisms?} |
95 | 334 |
|
122 | 335 |
\nn{Almost certainly we need a little more than the above axiom. |
336 |
More specifically, in order to bootstrap our way from the top dimension |
|
337 |
properties of identity morphisms to low dimensions, we need regular products, |
|
338 |
pinched products and even half-pinched products. |
|
142 | 339 |
I'm not sure what the best way to cleanly axiomatize the properties of these various |
340 |
products is. |
|
122 | 341 |
For the moment, I'll assume that all flavors of the product are at |
342 |
our disposal, and I'll plan on revising the axioms later.} |
|
343 |
||
128 | 344 |
\nn{current idea for fixing this: make the above axiom a ``preliminary version" |
345 |
(as we have already done with some of the other axioms), then state the official |
|
346 |
axiom for maps $\pi: E \to X$ which are almost fiber bundles. |
|
347 |
one option is to restrict E to be a (full/half/not)-pinched product (up to homeo). |
|
348 |
the alternative is to give some sort of local criterion for what's allowed. |
|
349 |
state a gluing axiom for decomps $E = E'\cup E''$ where all three are of the correct type. |
|
350 |
} |
|
351 |
||
95 | 352 |
All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |
353 |
The last axiom (below), concerning actions of |
|
354 |
homeomorphisms in the top dimension $n$, distinguishes the two cases. |
|
355 |
||
356 |
We start with the plain $n$-category case. |
|
357 |
||
267 | 358 |
\begin{axiom}[Isotopy invariance in dimension $n$]{\textup{\textbf{[preliminary]}}} |
187 | 359 |
Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
95 | 360 |
to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
187 | 361 |
Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$. |
267 | 362 |
\end{axiom} |
96 | 363 |
|
174 | 364 |
This axiom needs to be strengthened to force product morphisms to act as the identity. |
103 | 365 |
Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. |
96 | 366 |
Let $J$ be a 1-ball (interval). |
367 |
We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$. |
|
122 | 368 |
(Here we use the ``pinched" version of $Y\times J$. |
369 |
\nn{need notation for this}) |
|
96 | 370 |
We define a map |
371 |
\begin{eqnarray*} |
|
372 |
\psi_{Y,J}: \cC(X) &\to& \cC(X) \\ |
|
373 |
a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) . |
|
374 |
\end{eqnarray*} |
|
142 | 375 |
(See Figure \ref{glue-collar}.) |
189 | 376 |
\begin{figure}[!ht] |
377 |
\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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\foreach \i in {0, 1, 2} { |
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\node(\i) at ($\i*(\gap,0)$) [draw, circle through = {($\i*(\gap,0)+(\rad,0)$)}] {}; |
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\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[->] ($(0.east)+(0.75,0)$) -- ($(1.west)+(-0.2,0)$); |
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\draw (1-small) circle (\srad); |
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\foreach \theta in {90, 72, ..., -90} { |
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\draw[blue] (1) -- ($(1)+(\rad,0)+(\theta:\srad)$); |
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} |
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\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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\end{scope} |
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\end{tikzpicture} |
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\end{equation*} |
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\begin{equation*} |
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\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 | 418 |
\end{equation*} |
419 |
||
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\caption{Extended homeomorphism.}\label{glue-collar}\end{figure} |
|
174 | 421 |
We say that $\psi_{Y,J}$ is {\it extended isotopic} to the identity map. |
422 |
\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} |
|
97 | 425 |
\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} |
|
96 | 428 |
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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||
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\addtocounter{axiom}{-1} |
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\begin{axiom}{\textup{\textbf{[topological version]}} Extended isotopy invariance in dimension $n$} |
187 | 435 |
\label{axiom:extended-isotopies} |
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Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
|
174 | 437 |
to the identity on $\bd X$ and is extended isotopic (rel boundary) to the identity. |
187 | 438 |
Then $f$ acts trivially on $\cC(X)$. |
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\end{axiom} |
96 | 440 |
|
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\nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
|
94 | 442 |
|
97 | 443 |
\smallskip |
444 |
||
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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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||
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\addtocounter{axiom}{-1} |
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\begin{axiom}{\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$} |
187 | 451 |
For each $n$-ball $X$ and each $c\in \cC(\bd X)$ we have a map of chain complexes |
97 | 452 |
\[ |
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C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
|
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\] |
|
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Here $C_*$ means singular chains and $\Homeo_\bd(X)$ is the space of homeomorphisms of $X$ |
|
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which fix $\bd X$. |
|
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These action maps are required to be associative up to homotopy |
|
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\nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
|
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a diagram like the one in Proposition \ref{CHprop} commutes. |
97 | 460 |
\nn{repeat diagram here?} |
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\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} |
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\end{axiom} |
97 | 463 |
|
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We should strengthen the above axiom to apply to families of extended homeomorphisms. |
|
109 | 465 |
To do this we need to explain how extended homeomorphisms form a topological space. |
97 | 466 |
Roughly, the set of $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
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and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
|
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\nn{need to also say something about collaring homeomorphisms.} |
|
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\nn{this paragraph needs work.} |
|
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||
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Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
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into a plain $n$-category (enriched over graded groups). |
|
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\nn{say more here?} |
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In a different direction, if we enrich over topological spaces instead of chain complexes, |
97 | 475 |
we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
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instead of $C_*(\Homeo_\bd(X))$. |
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Taking singular chains converts such a space type $A_\infty$ $n$-category into a chain complex |
97 | 478 |
type $A_\infty$ $n$-category. |
479 |
||
99 | 480 |
\medskip |
97 | 481 |
|
99 | 482 |
The alert reader will have already noticed that our definition of (plain) $n$-category |
483 |
is extremely similar to our definition of topological fields. |
|
142 | 484 |
The main difference is that for the $n$-category definition we restrict our attention to balls |
99 | 485 |
(and their boundaries), while for fields we consider all manifolds. |
142 | 486 |
(A minor difference is that in the category definition we directly impose isotopy |
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invariance in dimension $n$, while in the fields definition we have non-isotopy-invariant fields |
|
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but then mod out by local relations which imply isotopy invariance.) |
|
99 | 489 |
Thus a system of fields determines an $n$-category simply by restricting our attention to |
490 |
balls. |
|
142 | 491 |
This $n$-category can be thought of as the local part of the fields. |
99 | 492 |
Conversely, given an $n$-category we can construct a system of fields via |
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a colimit construction; see \S \ref{ss:ncat_fields} below. |
99 | 494 |
|
142 | 495 |
%\nn{Next, say something about $A_\infty$ $n$-categories and ``homological" systems |
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%of fields. |
|
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%The universal (colimit) construction becomes our generalized definition of blob homology. |
|
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%Need to explain how it relates to the old definition.} |
|
97 | 499 |
|
95 | 500 |
\medskip |
501 |
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\subsection{Examples of $n$-categories} |
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\label{ss:ncat-examples} |
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101 | 505 |
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We now describe several classes of examples of $n$-categories satisfying our axioms. |
101 | 507 |
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\begin{example}[Maps to a space] |
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\rm |
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\label{ex:maps-to-a-space}% |
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Fix a `target space' $T$, any topological space. We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows. |
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For $X$ a $k$-ball with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of |
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all continuous maps from $X$ to $T$. |
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For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo |
196 | 515 |
homotopies fixed on $\bd X$. |
101 | 516 |
(Note that homotopy invariance implies isotopy invariance.) |
517 |
For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
|
518 |
be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. |
|
313 | 519 |
|
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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. Constructing a system of fields from $\pi_{\leq n}(T)$ recovers that example. |
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\end{example} |
101 | 522 |
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\begin{example}[Maps to a space, with a fiber] |
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\rm |
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\label{ex:maps-to-a-space-with-a-fiber}% |
196 | 526 |
We can modify the example above, by fixing a |
527 |
closed $m$-manifold $F$, and defining $\pi^{\times F}_{\leq n}(T)(X) = \Maps(X \times F \to T)$, otherwise leaving the definition in Example \ref{ex:maps-to-a-space} unchanged. Taking $F$ to be a point recovers the previous case. |
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\end{example} |
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529 |
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\begin{example}[Linearized, twisted, maps to a space] |
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\rm |
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\label{ex:linearized-maps-to-a-space}% |
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533 |
We can linearize Examples \ref{ex:maps-to-a-space} and \ref{ex:maps-to-a-space-with-a-fiber} as follows. |
101 | 534 |
Let $\alpha$ be an $(n{+}m{+}1)$-cocycle on $T$ with values in a ring $R$ |
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(have in mind the trivial cocycle). |
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For $X$ of dimension less than $n$ define $\pi^{\alpha, \times F}_{\leq n}(T)(X)$ as before, ignoring $\alpha$. |
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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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538 |
the $R$-module of finite linear combinations of continuous maps from $X\times F$ to $T$, |
101 | 539 |
modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy |
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$h: X\times F\times I \to T$, then $a = \alpha(h)b$. |
101 | 541 |
\nn{need to say something about fundamental classes, or choose $\alpha$ carefully} |
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\end{example} |
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543 |
|
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544 |
The next example is only intended to be illustrative, as we don't specify which definition of a `traditional $n$-category' we intend. Further, most of these definitions don't even have an agreed-upon notion of `strong duality', which we assume here. |
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\begin{example}[Traditional $n$-categories] |
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546 |
\rm |
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547 |
\label{ex:traditional-n-categories} |
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Given a `traditional $n$-category with strong duality' $C$ |
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549 |
define $\cC(X)$, for $X$ a $k$-ball with $k < n$, |
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to be the set of all $C$-labeled sub cell complexes of $X$ (c.f. \S \ref{sec:fields}). |
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For $X$ an $n$-ball and $c\in \cC(\bd X)$, define $\cC(X)$ to finite linear |
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552 |
combinations of $C$-labeled sub cell complexes of $X$ |
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modulo the kernel of the evaluation map. |
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554 |
Define a product morphism $a\times D$, for $D$ an $m$-ball, to be the product of the cell complex of $a$ with $D$, |
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with each cell labelled by the $m$-th iterated identity morphism of the corresponding cell for $a$. |
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556 |
More generally, start with an $n{+}m$-category $C$ and a closed $m$-manifold $F$. |
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Define $\cC(X)$, for $\dim(X) < n$, |
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to be the set of all $C$-labeled sub cell complexes of $X\times F$. |
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559 |
Define $\cC(X; c)$, for $X$ an $n$-ball, |
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to be the dual Hilbert space $A(X\times F; c)$. |
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\nn{refer elsewhere for details?} |
313 | 562 |
|
563 |
||
564 |
Recall we described a system of fields and local relations based on a `traditional $n$-category' $C$ in Example \ref{ex:traditional-n-categories(fields)} above. Constructing a system of fields from $\cC$ recovers that example. |
|
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\end{example} |
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566 |
|
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Finally, we describe a version of the bordism $n$-category suitable to our definitions. |
204 | 568 |
|
569 |
\nn{should also include example of ncats coming from TQFTs, or refer ahead to where we discuss that example} |
|
570 |
||
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\newcommand{\Bord}{\operatorname{Bord}} |
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\begin{example}[The bordism $n$-category, plain version] |
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573 |
\rm |
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\label{ex:bordism-category} |
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For a $k$-ball $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional |
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submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse |
196 | 577 |
to $\bd X$. |
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|
578 |
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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579 |
we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism |
196 | 580 |
$W \to W'$ which restricts to the identity on the boundary. |
191
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581 |
\end{example} |
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582 |
|
196 | 583 |
%\nn{the next example might be an unnecessary distraction. consider deleting it.} |
101 | 584 |
|
196 | 585 |
%\begin{example}[Variation on the above examples] |
586 |
%We could allow $F$ to have boundary and specify boundary conditions on $X\times \bd F$, |
|
587 |
%for example product boundary conditions or take the union over all boundary conditions. |
|
588 |
%%\nn{maybe should not emphasize this case, since it's ``better" in some sense |
|
589 |
%%to think of these guys as affording a representation |
|
590 |
%%of the $n{+}1$-category associated to $\bd F$.} |
|
591 |
%\end{example} |
|
101 | 592 |
|
593 |
||
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594 |
%We have two main examples of $A_\infty$ $n$-categories, coming from maps to a target space and from the blob complex. |
101 | 595 |
|
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596 |
\begin{example}[Chains of maps to a space] |
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597 |
\rm |
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598 |
\label{ex:chains-of-maps-to-a-space} |
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599 |
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 $T$. |
310
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600 |
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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601 |
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 |
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602 |
$$C_*(\Maps_c(X\times F \to T)),$$ where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
101 | 603 |
and $C_*$ denotes singular chains. |
211 | 604 |
\nn{maybe should also mention version where we enrich over spaces rather than chain complexes} |
190
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605 |
\end{example} |
101 | 606 |
|
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607 |
See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to homotopy the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
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|
608 |
|
279 | 609 |
\begin{example}[Blob complexes of balls (with a fiber)] |
610 |
\rm |
|
611 |
\label{ex:blob-complexes-of-balls} |
|
291 | 612 |
Fix an $n-k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$. |
613 |
We will define an $A_\infty$ $k$-category $\cC$. |
|
310
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614 |
When $X$ is a $m$-ball, with $m<k$, define $\cC(X) = \cE(X\times F)$. |
291 | 615 |
When $X$ is an $k$-ball, |
279 | 616 |
define $\cC(X; c) = \bc^\cE_*(X\times F; c)$ |
617 |
where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
|
618 |
\end{example} |
|
101 | 619 |
|
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|
620 |
This example will be essential for Theorem \ref{product_thm} below, which allows us to compute the blob complex of a product. Notice that with $F$ a point, the above example is a construction turning a topological $n$-category into an $A_\infty$ $n$-category. We think of this as providing a `free resolution' of the topological $n$-category. \todo{Say more here!} In fact, there is also a trivial way to do this: we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, and take $\CD{B}$ to act trivially. |
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621 |
|
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622 |
Be careful that the `free resolution' of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. It's easy to see that with $n=0$, the corresponding system of fields is just linear combinations of connected components of $T$, and the local relations are trivial. There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
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623 |
|
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624 |
\begin{example}[The bordism $n$-category, $A_\infty$ version] |
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625 |
\rm |
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626 |
\label{ex:bordism-category-ainf} |
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627 |
blah blah \nn{to do...} |
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|
628 |
\end{example} |
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629 |
|
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630 |
|
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|
631 |
\begin{example}[$E_n$ algebras] |
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632 |
\rm |
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|
633 |
\label{ex:e-n-alg} |
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634 |
Let $\cE\cB_n$ be the operad of smooth embeddings of $k$ (little) |
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635 |
copies of the standard $n$-ball $B^n$ into another (big) copy of $B^n$. |
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636 |
$\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad. |
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637 |
(By shrining the little balls, we see that both are homotopic to the space of $k$ framed points |
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638 |
in $B^n$.) |
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639 |
|
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640 |
Let $A$ be an $\cE\cB_n$-algebra. |
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641 |
We will define an $A_\infty$ $n$-category $\cC^A$. |
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642 |
\nn{...} |
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643 |
\end{example} |
95 | 644 |
|
108 | 645 |
|
646 |
||
647 |
||
648 |
||
649 |
||
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650 |
%\subsection{From $n$-categories to systems of fields} |
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651 |
\subsection{From balls to manifolds} |
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652 |
\label{ss:ncat_fields} \label{ss:ncat-coend} |
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653 |
In this section we describe how to extend an $n$-category as described above (of either the plain or $A_\infty$ variety) to a system of fields. |
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654 |
That is, we show that functors $\cC_k$ satisfying the axioms above have a canonical extension |
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655 |
from $k$-balls to arbitrary $k$-manifolds. |
204 | 656 |
In the case of plain $n$-categories, this is just the usual construction of a TQFT |
657 |
from an $n$-category. |
|
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658 |
For $A_\infty$ $n$-categories, this gives an alternate (and |
204 | 659 |
somewhat more canonical/tautological) construction of the blob complex. |
660 |
\nn{though from this point of view it seems more natural to just add some |
|
661 |
adjective to ``TQFT" rather than coining a completely new term like ``blob complex".} |
|
108 | 662 |
|
197 | 663 |
We will first define the `cell-decomposition' poset $\cJ(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
664 |
An $n$-category $\cC$ provides a functor from this poset to the category of sets, and we will define $\cC(W)$ as a suitable colimit (or homotopy colimit in the $A_\infty$ case) of this functor. |
|
665 |
We'll later give a more explicit description of this colimit. In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-manifolds with boundary data), then the resulting system of fields 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 | 666 |
|
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667 |
\begin{defn} |
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668 |
Say that a `permissible decomposition' of $W$ is a cell decomposition |
108 | 669 |
\[ |
670 |
W = \bigcup_a X_a , |
|
671 |
\] |
|
142 | 672 |
where each closed top-dimensional cell $X_a$ is an embedded $k$-ball. |
191
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673 |
|
108 | 674 |
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
191
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675 |
of $y$, or write $x \le y$, if each $k$-ball of $y$ is a union of $k$-balls of $x$. |
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676 |
|
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677 |
The category $\cJ(W)$ has objects the permissible decompositions of $W$, and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
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678 |
See Figure \ref{partofJfig} for an example. |
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679 |
\end{defn} |
119 | 680 |
|
681 |
\begin{figure}[!ht] |
|
682 |
\begin{equation*} |
|
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683 |
\mathfig{.63}{ncat/zz2} |
119 | 684 |
\end{equation*} |
685 |
\caption{A small part of $\cJ(W)$} |
|
686 |
\label{partofJfig} |
|
687 |
\end{figure} |
|
688 |
||
108 | 689 |
|
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690 |
|
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691 |
An $n$-category $\cC$ determines |
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692 |
a functor $\psi_{\cC;W}$ from $\cJ(W)$ to the category of sets |
108 | 693 |
(possibly with additional structure if $k=n$). |
197 | 694 |
Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
695 |
and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries |
|
696 |
are splittable along this decomposition. |
|
697 |
%For a $k$-cell $X$ in a cell composition of $W$, we can consider the `splittable fields' $\cC(X)_{\bdy X}$, the subset of $\cC(X)$ consisting of fields which are splittable with respect to each boundary $k-1$-cell. |
|
108 | 698 |
|
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699 |
\begin{defn} |
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700 |
Define the functor $\psi_{\cC;W} : \cJ(W) \to \Set$ as follows. |
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701 |
For a decomposition $x = \bigcup_a X_a$ in $\cJ(W)$, $\psi_{\cC;W}(x)$ is the subset |
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702 |
\begin{equation} |
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703 |
\label{eq:psi-C} |
197 | 704 |
\psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl |
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705 |
\end{equation} |
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706 |
where the restrictions to the various pieces of shared boundaries amongst the cells |
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707 |
$X_a$ 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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708 |
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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709 |
\end{defn} |
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710 |
|
197 | 711 |
When the $n$-category $\cC$ is enriched in some monoidal category $(A,\boxtimes)$, and $W$ is a |
712 |
closed $n$-manifold, the functor $\psi_{\cC;W}$ has target $A$ and |
|
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713 |
we replace the cartesian product of sets appearing in Equation \eqref{eq:psi-C} with the monoidal product $\boxtimes$. (Moreover, $\psi_{\cC;W}(x)$ might be a subobject, rather than a subset, of the product.) |
197 | 714 |
Similar things are true if $W$ is an $n$-manifold with non-empty boundary and we |
715 |
fix a field on $\bd W$ |
|
716 |
(i.e. fix an element of the colimit associated to $\bd W$). |
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717 |
|
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718 |
Finally, we construct $\cC(W)$ as the appropriate colimit of $\psi_{\cC;W}$. |
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719 |
|
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720 |
\begin{defn}[System of fields functor] |
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721 |
If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cC(W)$ is the usual colimit of the functor $\psi_{\cC;W}$. |
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722 |
That is, for each decomposition $x$ there is a map |
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|
723 |
$\psi_{\cC;W}(x)\to \cC(W)$, these maps are compatible with the refinement maps |
108 | 724 |
above, and $\cC(W)$ is universal with respect to these properties. |
191
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|
725 |
\end{defn} |
112 | 726 |
|
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|
727 |
\begin{defn}[System of fields functor, $A_\infty$ case] |
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|
728 |
When $\cC$ is an $A_\infty$ $n$-category, $\cC(W)$ for $W$ a $k$-manifold with $k < n$ is defined as above, as the colimit of $\psi_{\cC;W}$. When $W$ is an $n$-manifold, the chain complex $\cC(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$. |
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|
729 |
\end{defn} |
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|
730 |
|
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|
731 |
We can specify boundary data $c \in \cC(\bdy W)$, and define functors $\psi_{\cC;W,c}$ with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$. |
111 | 732 |
|
197 | 733 |
We now give a more concrete description of the colimit in each case. If $\cC$ is enriched over vector spaces, and $W$ is an $n$-manifold, we can take the vector space $\cC(W,c)$ to be the direct sum over all permissible decompositions of $W$ |
191
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|
734 |
\begin{equation*} |
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|
735 |
\cC(W,c) = \left( \bigoplus_x \psi_{\cC;W,c}(x)\right) \big/ K |
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|
736 |
\end{equation*} |
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|
737 |
where $K$ is the vector space spanned by elements $a - g(a)$, with |
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|
738 |
$a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
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|
739 |
\to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
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|
740 |
|
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|
741 |
In the $A_\infty$ case, enriched over chain complexes, the concrete description of the homotopy colimit |
197 | 742 |
is more involved. |
142 | 743 |
%\nn{should probably rewrite this to be compatible with some standard reference} |
191
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744 |
Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$. |
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745 |
Such sequences (for all $m$) form a simplicial set in $\cJ(W)$. |
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|
746 |
Define $V$ as a vector space via |
112 | 747 |
\[ |
191
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|
748 |
V = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
112 | 749 |
\] |
198 | 750 |
where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$. (Our homological conventions are non-standard: if a complex $U$ is concentrated in degree $0$, the complex $U[m]$ is concentrated in degree $m$.) |
191
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|
751 |
We endow $V$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ |
112 | 752 |
summands plus another term using the differential of the simplicial set of $m$-sequences. |
753 |
More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
|
754 |
summand of $V$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
|
755 |
\[ |
|
191
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|
756 |
\bd (a, \bar{x}) = (\bd a, \bar{x}) + (-1)^{\deg{a}} (g(a), d_0(\bar{x})) + (-1)^{\deg{a}} \sum_{j=1}^k (-1)^{j} (a, d_j(\bar{x})) , |
112 | 757 |
\] |
758 |
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 | 759 |
is the usual gluing map coming from the antirefinement $x_0 \le x_1$. |
112 | 760 |
\nn{need to say this better} |
761 |
\nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which |
|
762 |
combine only two balls at a time; for $n=1$ this version will lead to usual definition |
|
763 |
of $A_\infty$ category} |
|
108 | 764 |
|
113 | 765 |
We will call $m$ the filtration degree of the complex. |
766 |
We can think of this construction as starting with a disjoint copy of a complex for each |
|
767 |
permissible decomposition (filtration degree 0). |
|
768 |
Then we glue these together with mapping cylinders coming from gluing maps |
|
769 |
(filtration degree 1). |
|
267 | 770 |
Then we kill the extra homology we just introduced with mapping cylinders between the mapping cylinders (filtration degree 2), and so on. |
113 | 771 |
|
108 | 772 |
$\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
773 |
||
774 |
It is easy to see that |
|
775 |
there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps |
|
776 |
comprise a natural transformation of functors. |
|
777 |
||
778 |
\nn{need to finish explaining why we have a system of fields; |
|
779 |
need to say more about ``homological" fields? |
|
780 |
(actions of homeomorphisms); |
|
781 |
define $k$-cat $\cC(\cdot\times W)$} |
|
782 |
||
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|
783 |
\nn{need to revise stuff below, since we no longer have the sphere axiom} |
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|
784 |
|
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|
785 |
Recall that Axiom \ref{axiom:spheres} for an $n$-category provided functors $\cC$ from $k$-spheres to sets for $0 \leq k < n$. We claim now that these functors automatically agree with the colimits we have associated to spheres in this section. \todo{} \todo{In fact, we probably should do this for balls as well!} For the remainder of this section we will write $\underrightarrow{\cC}(W)$ for the colimit associated to an arbitary manifold $W$, to distinguish it, in the case that $W$ is a ball or a sphere, from $\cC(W)$, which is part of the definition of the $n$-category. After the next three lemmas, there will be no further need for this notational distinction. |
267 | 786 |
|
787 |
\begin{lem} |
|
788 |
For a $k$-ball or $k$-sphere $W$, with $0\leq k < n$, $$\underrightarrow{\cC}(W) = \cC(W).$$ |
|
789 |
\end{lem} |
|
790 |
||
791 |
\begin{lem} |
|
792 |
For a topological $n$-category $\cC$, and an $n$-ball $B$, $$\underrightarrow{\cC}(B) = \cC(B).$$ |
|
793 |
\end{lem} |
|
794 |
||
795 |
\begin{lem} |
|
796 |
For an $A_\infty$ $n$-category $\cC$, and an $n$-ball $B$, $$\underrightarrow{\cC}(B) \quism \cC(B).$$ |
|
797 |
\end{lem} |
|
108 | 798 |
|
799 |
||
800 |
\subsection{Modules} |
|
95 | 801 |
|
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|
802 |
Next we define plain and $A_\infty$ $n$-category modules. |
199 | 803 |
The definition will be very similar to that of $n$-categories, |
804 |
but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
|
109 | 805 |
\nn{** need to make sure all revisions of $n$-cat def are also made to module def.} |
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|
806 |
\nn{in particular, need to to get rid of the ``hemisphere axiom"} |
198 | 807 |
%\nn{should they be called $n$-modules instead of just modules? probably not, but worth considering.} |
808 |
||
104 | 809 |
Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
102 | 810 |
in the context of an $m{+}1$-dimensional TQFT. |
811 |
Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
|
812 |
This will be explained in more detail as we present the axioms. |
|
813 |
||
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|
814 |
\nn{should also develop $\pi_{\le n}(T, S)$ as a module for $\pi_{\le n}(T)$, where $S\sub T$.} |
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|
815 |
|
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|
816 |
Throughout, we fix an $n$-category $\cC$. For all but one axiom, it doesn't matter whether $\cC$ is a topological $n$-category or an $A_\infty$ $n$-category. We state the final axiom, on actions of homeomorphisms, differently in the two cases. |
102 | 817 |
|
818 |
Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair |
|
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|
819 |
$$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$ |
102 | 820 |
We call $B$ the ball and $N$ the marking. |
821 |
A homeomorphism between marked $k$-balls is a homeomorphism of balls which |
|
822 |
restricts to a homeomorphism of markings. |
|
823 |
||
199 | 824 |
\mmpar{Module axiom 1}{Module morphisms} |
102 | 825 |
{For each $0 \le k \le n$, we have a functor $\cM_k$ from |
826 |
the category of marked $k$-balls and |
|
827 |
homeomorphisms to the category of sets and bijections.} |
|
828 |
||
829 |
(As with $n$-categories, we will usually omit the subscript $k$.) |
|
830 |
||
104 | 831 |
For example, let $\cD$ be the $m{+}1$-dimensional TQFT which assigns to a $k$-manifold $N$ the set |
832 |
of maps from $N$ to $T$, modulo homotopy (and possibly linearized) if $k=m$. |
|
833 |
Let $W$ be an $(m{-}n{+}1)$-dimensional manifold with boundary. |
|
834 |
Let $\cC$ be the $n$-category with $\cC(X) \deq \cD(X\times \bd W)$. |
|
835 |
Let $\cM(B, N) \deq \cD((B\times \bd W)\cup (N\times W))$. |
|
836 |
(The union is along $N\times \bd W$.) |
|
110 | 837 |
(If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be |
838 |
the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.) |
|
102 | 839 |
|
182 | 840 |
\begin{figure}[!ht] |
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|
841 |
$$\mathfig{.8}{ncat/boundary-collar}$$ |
182 | 842 |
\caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure} |
843 |
||
103 | 844 |
Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
845 |
Call such a thing a {marked $k{-}1$-hemisphere}. |
|
102 | 846 |
|
199 | 847 |
\mmpar{Module axiom 2}{Module boundaries (hemispheres)} |
102 | 848 |
{For each $0 \le k \le n-1$, we have a functor $\cM_k$ from |
104 | 849 |
the category of marked $k$-hemispheres and |
102 | 850 |
homeomorphisms to the category of sets and bijections.} |
851 |
||
104 | 852 |
In our example, let $\cM(H) \deq \cD(H\times\bd W \cup \bd H\times W)$. |
853 |
||
199 | 854 |
\mmpar{Module axiom 3}{Module boundaries (maps)} |
102 | 855 |
{For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cM(\bd M)$. |
856 |
These maps, for various $M$, comprise a natural transformation of functors.} |
|
857 |
||
110 | 858 |
Given $c\in\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
102 | 859 |
|
860 |
If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
|
861 |
then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$ |
|
862 |
and $c\in \cC(\bd M)$. |
|
863 |
||
199 | 864 |
\mmpar{Module axiom 4}{Boundary from domain and range} |
102 | 865 |
{Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k$-hemisphere ($0\le k\le n-1$), |
104 | 866 |
$M_i$ is a marked $k$-ball, and $E = M_1\cap M_2$ is a marked $k{-}1$-hemisphere. |
867 |
Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the |
|
868 |
two maps $\bd: \cM(M_i)\to \cM(E)$. |
|
102 | 869 |
Then (axiom) we have an injective map |
870 |
\[ |
|
199 | 871 |
\gl_E : \cM(M_1) \times_{\cM(E)} \cM(M_2) \hookrightarrow \cM(H) |
102 | 872 |
\] |
873 |
which is natural with respect to the actions of homeomorphisms.} |
|
874 |
||
110 | 875 |
Let $\cM(H)_E$ denote the image of $\gl_E$. |
876 |
We will refer to elements of $\cM(H)_E$ as ``splittable along $E$" or ``transverse to $E$". |
|
877 |
||
878 |
||
199 | 879 |
\mmpar{Module axiom 5}{Module to category restrictions} |
103 | 880 |
{For each marked $k$-hemisphere $H$ there is a restriction map |
881 |
$\cM(H)\to \cC(H)$. |
|
882 |
($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) |
|
883 |
These maps comprise a natural transformation of functors.} |
|
102 | 884 |
|
103 | 885 |
Note that combining the various boundary and restriction maps above |
110 | 886 |
(for both modules and $n$-categories) |
103 | 887 |
we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$ |
888 |
a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$. |
|
110 | 889 |
The subset is the subset of morphisms which are appropriately splittable (transverse to the |
890 |
cutting submanifolds). |
|
103 | 891 |
This fact will be used below. |
102 | 892 |
|
104 | 893 |
In our example, the various restriction and gluing maps above come from |
894 |
restricting and gluing maps into $T$. |
|
895 |
||
896 |
We require two sorts of composition (gluing) for modules, corresponding to two ways |
|
103 | 897 |
of splitting a marked $k$-ball into two (marked or plain) $k$-balls. |
119 | 898 |
(See Figure \ref{zzz3}.) |
103 | 899 |
|
119 | 900 |
\begin{figure}[!ht] |
901 |
\begin{equation*} |
|
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|
902 |
\mathfig{.4}{ncat/zz3} |
119 | 903 |
\end{equation*} |
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|
904 |
\caption{Module composition (top); $n$-category action (bottom).} |
119 | 905 |
\label{zzz3} |
906 |
\end{figure} |
|
907 |
||
908 |
First, we can compose two module morphisms to get another module morphism. |
|
103 | 909 |
|
200 | 910 |
\mmpar{Module axiom 6}{Module composition} |
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|
911 |
{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 | 912 |
and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball. |
913 |
Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere. |
|
914 |
Note that each of $M$, $M_1$ and $M_2$ has its boundary split into two marked $k{-}1$-balls by $E$. |
|
915 |
We have restriction (domain or range) maps $\cM(M_i)_E \to \cM(Y)$. |
|
916 |
Let $\cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E$ denote the fibered product of these two maps. |
|
917 |
Then (axiom) we have a map |
|
918 |
\[ |
|
919 |
\gl_Y : \cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E \to \cM(M)_E |
|
920 |
\] |
|
921 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
922 |
to the intersection of the boundaries of $M$ and $M_i$. |
|
923 |
If $k < n$ we require that $\gl_Y$ is injective. |
|
924 |
(For $k=n$, see below.)} |
|
925 |
||
119 | 926 |
|
927 |
||
103 | 928 |
Second, we can compose an $n$-category morphism with a module morphism to get another |
929 |
module morphism. |
|
930 |
We'll call this the action map to distinguish it from the other kind of composition. |
|
931 |
||
200 | 932 |
\mmpar{Module axiom 7}{$n$-category action} |
103 | 933 |
{Let $M = X \cup_Y M'$, where $M$ and $M'$ are marked $k$-balls ($0\le k\le n$), |
934 |
$X$ is a plain $k$-ball, |
|
935 |
and $Y = X\cap M'$ is a $k{-}1$-ball. |
|
936 |
Let $E = \bd Y$, which is a $k{-}2$-sphere. |
|
937 |
We have restriction maps $\cM(M')_E \to \cC(Y)$ and $\cC(X)_E\to \cC(Y)$. |
|
938 |
Let $\cC(X)_E \times_{\cC(Y)} \cM(M')_E$ denote the fibered product of these two maps. |
|
939 |
Then (axiom) we have a map |
|
940 |
\[ |
|
941 |
\gl_Y :\cC(X)_E \times_{\cC(Y)} \cM(M')_E \to \cM(M)_E |
|
942 |
\] |
|
943 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
944 |
to the intersection of the boundaries of $X$ and $M'$. |
|
945 |
If $k < n$ we require that $\gl_Y$ is injective. |
|
946 |
(For $k=n$, see below.)} |
|
947 |
||
200 | 948 |
\mmpar{Module axiom 8}{Strict associativity} |
103 | 949 |
{The composition and action maps above are strictly associative.} |
950 |
||
110 | 951 |
Note that the above associativity axiom applies to mixtures of module composition, |
952 |
action maps and $n$-category composition. |
|
119 | 953 |
See Figure \ref{zzz1b}. |
954 |
||
955 |
\begin{figure}[!ht] |
|
956 |
\begin{equation*} |
|
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957 |
\mathfig{0.49}{ncat/zz0} \mathfig{0.49}{ncat/zz1} |
119 | 958 |
\end{equation*} |
959 |
\caption{Two examples of mixed associativity} |
|
960 |
\label{zzz1b} |
|
961 |
\end{figure} |
|
962 |
||
110 | 963 |
|
964 |
The above three axioms are equivalent to the following axiom, |
|
103 | 965 |
which we state in slightly vague form. |
966 |
\nn{need figure for this} |
|
967 |
||
968 |
\xxpar{Module multi-composition:} |
|
969 |
{Given any decomposition |
|
970 |
\[ |
|
971 |
M = X_1 \cup\cdots\cup X_p \cup M_1\cup\cdots\cup M_q |
|
972 |
\] |
|
973 |
of a marked $k$-ball $M$ |
|
974 |
into small (marked and plain) $k$-balls $M_i$ and $X_j$, there is a |
|
975 |
map from an appropriate subset (like a fibered product) |
|
976 |
of |
|
977 |
\[ |
|
978 |
\cC(X_1)\times\cdots\times\cC(X_p) \times \cM(M_1)\times\cdots\times\cM(M_q) |
|
979 |
\] |
|
980 |
to $\cM(M)$, |
|
981 |
and these various multifold composition maps satisfy an |
|
982 |
operad-type strict associativity condition.} |
|
983 |
||
984 |
(The above operad-like structure is analogous to the swiss cheese operad |
|
146 | 985 |
\cite{MR1718089}.) |
200 | 986 |
%\nn{need to double-check that this is true.} |
103 | 987 |
|
200 | 988 |
\mmpar{Module axiom 9}{Product/identity morphisms} |
103 | 989 |
{Let $M$ be a marked $k$-ball and $D$ be a plain $m$-ball, with $k+m \le n$. |
990 |
Then we have a map $\cM(M)\to \cM(M\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cM(M)$. |
|
991 |
If $f:M\to M'$ and $\tilde{f}:M\times D \to M'\times D'$ are maps such that the diagram |
|
992 |
\[ \xymatrix{ |
|
993 |
M\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & M'\times D' \ar[d]^{\pi} \\ |
|
994 |
M \ar[r]^{f} & M' |
|
995 |
} \] |
|
996 |
commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.} |
|
997 |
||
111 | 998 |
\nn{Need to add compatibility with various things, as in the n-cat version of this axiom above.} |
103 | 999 |
|
200 | 1000 |
\nn{postpone finalizing the above axiom until the n-cat version is finalized} |
110 | 1001 |
|
103 | 1002 |
There are two alternatives for the next axiom, according whether we are defining |
1003 |
modules for plain $n$-categories or $A_\infty$ $n$-categories. |
|
1004 |
In the plain case we require |
|
1005 |
||
200 | 1006 |
\mmpar{Module axiom 10a}{Extended isotopy invariance in dimension $n$} |
103 | 1007 |
{Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts |
175 | 1008 |
to the identity on $\bd M$ and is extended isotopic (rel boundary) to the identity. |
103 | 1009 |
Then $f$ acts trivially on $\cM(M)$.} |
1010 |
||
1011 |
\nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
|
1012 |
||
1013 |
We emphasize that the $\bd M$ above means boundary in the marked $k$-ball sense. |
|
1014 |
In other words, if $M = (B, N)$ then we require only that isotopies are fixed |
|
1015 |
on $\bd B \setmin N$. |
|
1016 |
||
1017 |
For $A_\infty$ modules we require |
|
1018 |
||
200 | 1019 |
\mmpar{Module axiom 10b}{Families of homeomorphisms act} |
103 | 1020 |
{For each marked $n$-ball $M$ and each $c\in \cM(\bd M)$ we have a map of chain complexes |
1021 |
\[ |
|
1022 |
C_*(\Homeo_\bd(M))\ot \cM(M; c) \to \cM(M; c) . |
|
1023 |
\] |
|
1024 |
Here $C_*$ means singular chains and $\Homeo_\bd(M)$ is the space of homeomorphisms of $M$ |
|
1025 |
which fix $\bd M$. |
|
1026 |
These action maps are required to be associative up to homotopy |
|
1027 |
\nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
|
236 | 1028 |
a diagram like the one in Proposition \ref{CHprop} commutes. |
103 | 1029 |
\nn{repeat diagram here?} |
1030 |
\nn{restate this with $\Homeo(M\to M')$? what about boundary fixing property?}} |
|
1031 |
||
1032 |
\medskip |
|
102 | 1033 |
|
104 | 1034 |
Note that the above axioms imply that an $n$-category module has the structure |
1035 |
of an $n{-}1$-category. |
|
1036 |
More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$, |
|
1037 |
where $X$ is a $k$-ball or $k{-}1$-sphere and in the product $X\times J$ we pinch |
|
1038 |
above the non-marked boundary component of $J$. |
|
200 | 1039 |
(More specifically, we collapse $X\times P$ to a single point, where |
1040 |
$P$ is the non-marked boundary component of $J$.) |
|
1041 |
\nn{give figure for this?} |
|
104 | 1042 |
Then $\cE$ has the structure of an $n{-}1$-category. |
102 | 1043 |
|
105 | 1044 |
All marked $k$-balls are homeomorphic, unless $k = 1$ and our manifolds |
1045 |
are oriented or Spin (but not unoriented or $\text{Pin}_\pm$). |
|
1046 |
In this case ($k=1$ and oriented or Spin), there are two types |
|
1047 |
of marked 1-balls, call them left-marked and right-marked, |
|
1048 |
and hence there are two types of modules, call them right modules and left modules. |
|
1049 |
In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$), |
|
1050 |
there is no left/right module distinction. |
|
1051 |
||
130 | 1052 |
\medskip |
1053 |
||
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1054 |
We now give some examples of modules over topological and $A_\infty$ $n$-categories. |
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1055 |
|
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1056 |
\begin{example}[Examples from TQFTs] |
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1057 |
\todo{} |
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1058 |
\end{example} |
108 | 1059 |
|
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1060 |
\begin{example} |
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1061 |
Suppose $S$ is a topological space, with a subspace $T$. We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$ for $k<n$ the set $\pi_{\leq n}(S,T)(B,N)$ consists of all continuous maps of pairs $(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all such maps modulo homotopies fixed on $\bdy B \setminus N$. This is a module over the fundamental $n$-category $\pi_{\leq n}(S)$ of $S$, from Example \ref{ex:maps-to-a-space}. Modifications corresponding to Examples \ref{ex:maps-to-a-space-with-a-fiber} and \ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains. |
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|
1062 |
\end{example} |
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|
1063 |
|
108 | 1064 |
\subsection{Modules as boundary labels} |
112 | 1065 |
\label{moddecss} |
108 | 1066 |
|
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1067 |
Fix a topological $n$-category or $A_\infty$ $n$-category $\cC$. Let $W$ be a $k$-manifold ($k\le n$), |
143 | 1068 |
let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$, |
1069 |
and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$. |
|
1070 |
||
1071 |
%Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), |
|
1072 |
%and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary |
|
1073 |
%component $\bd_i W$ of $W$. |
|
1074 |
%(More generally, each $\cN_i$ could label some codimension zero submanifold of $\bd W$.) |
|
108 | 1075 |
|
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|
1076 |
We will define a set $\cC(W, \cN)$ using a colimit construction similar to the one appearing in \S \ref{ss:ncat_fields} above. |
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|
1077 |
(If $k = n$ and our $n$-categories are enriched, then |
108 | 1078 |
$\cC(W, \cN)$ will have additional structure; see below.) |
1079 |
||
1080 |
Define a permissible decomposition of $W$ to be a decomposition |
|
1081 |
\[ |
|
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|
1082 |
W = \left(\bigcup_a X_a\right) \cup \left(\bigcup_{i,b} M_{ib}\right) , |
108 | 1083 |
\] |
1084 |
where each $X_a$ is a plain $k$-ball (disjoint from $\bd W$) and |
|
1085 |
each $M_{ib}$ is a marked $k$-ball intersecting $\bd_i W$, |
|
143 | 1086 |
with $M_{ib}\cap Y_i$ being the marking. |
1087 |
(See Figure \ref{mblabel}.) |
|
1088 |
\begin{figure}[!ht]\begin{equation*} |
|
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|
1089 |
\mathfig{.4}{ncat/mblabel} |
143 | 1090 |
\end{equation*}\caption{A permissible decomposition of a manifold |
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|
1091 |
whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$. Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel}\end{figure} |
108 | 1092 |
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
1093 |
of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
|
1094 |
This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
|
1095 |
(The objects of $\cJ(D)$ are permissible decompositions of $W$, and there is a unique |
|
1096 |
morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
|
1097 |
||
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|
1098 |
The collection of modules $\cN$ determines |
108 | 1099 |
a functor $\psi_\cN$ from $\cJ(W)$ to the category of sets |
1100 |
(possibly with additional structure if $k=n$). |
|
1101 |
For a decomposition $x = (X_a, M_{ib})$ in $\cJ(W)$, define $\psi_\cN(x)$ to be the subset |
|
1102 |
\[ |
|
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|
1103 |
\psi_\cN(x) \sub \left(\prod_a \cC(X_a)\right) \times \left(\prod_{ib} \cN_i(M_{ib})\right) |
108 | 1104 |
\] |
1105 |
such that the restrictions to the various pieces of shared boundaries amongst the |
|
1106 |
$X_a$ and $M_{ib}$ all agree. |
|
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|
1107 |
(That is, the fibered product over the boundary maps.) |
108 | 1108 |
If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$ |
1109 |
via the gluing (composition or action) maps from $\cC$ and the $\cN_i$. |
|
1110 |
||
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|
1111 |
We now define the set $\cC(W, \cN)$ to be the colimit of the functor $\psi_\cN$. |
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|
1112 |
(As usual, if $k=n$ and we are in the $A_\infty$ case, then ``colimit" means |
143 | 1113 |
homotopy colimit.) |
108 | 1114 |
|
143 | 1115 |
If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define |
1116 |
$\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold |
|
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|
1117 |
$D\times Y_i \sub \bd(D\times W)$. It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$ |
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|
1118 |
has the structure of an $n{-}k$-category, which we call $\cT(W, \cN)(D)$. |
144 | 1119 |
|
1120 |
\medskip |
|
1121 |
||
1122 |
||
1123 |
We will use a simple special case of the above |
|
1124 |
construction to define tensor products |
|
1125 |
of modules. |
|
1126 |
Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$. |
|
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|
1127 |
(If $k=1$ and our manifolds are oriented, then one should be |
144 | 1128 |
a left module and the other a right module.) |
1129 |
Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$. |
|
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1130 |
Define the tensor product $\cM_1 \tensor \cM_2$ to be the |
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|
1131 |
$n{-}1$-category $\cT(J, \{\cM_1, \cM_2\})$. This of course depends (functorially) |
144 | 1132 |
on the choice of 1-ball $J$. |
105 | 1133 |
|
144 | 1134 |
We will define a more general self tensor product (categorified coend) below. |
1135 |
||
1136 |
%\nn{what about self tensor products /coends ?} |
|
105 | 1137 |
|
108 | 1138 |
\nn{maybe ``tensor product" is not the best name?} |
1139 |
||
144 | 1140 |
%\nn{start with (less general) tensor products; maybe change this later} |
106 | 1141 |
|
107 | 1142 |
|
1143 |
||
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1144 |
|
291 | 1145 |
\subsection{Morphisms of $A_\infty$ $1$-category modules} |
288 | 1146 |
\label{ss:module-morphisms} |
258
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1147 |
|
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1148 |
In order to state and prove our version of the higher dimensional Deligne conjecture |
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|
1149 |
(Section \ref{sec:deligne}), |
291 | 1150 |
we need to define morphisms of $A_\infty$ $1$-category modules and establish |
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|
1151 |
some of their elementary properties. |
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1152 |
|
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|
1153 |
To motivate the definitions which follow, consider algebras $A$ and $B$, right modules $X_B$ and $Z_A$ and a bimodule $\leftidx{_B}{Y}{_A}$, and the familiar adjunction |
258
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1154 |
\begin{eqnarray*} |
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|
1155 |
\hom_A(X_B\ot {_BY_A} \to Z_A) &\cong& \hom_B(X_B \to \hom_A( {_BY_A} \to Z_A)) \\ |
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1156 |
f &\mapsto& [x \mapsto f(x\ot -)] \\ |
279 | 1157 |
{}[x\ot y \mapsto g(x)(y)] & \mapsfrom & g . |
258
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1158 |
\end{eqnarray*} |
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|
1159 |
If $A$ and $Z_A$ are both the ground field $\k$, this simplifies to |
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|
1160 |
\[ |
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|
1161 |
(X_B\ot {_BY})^* \cong \hom_B(X_B \to (_BY)^*) . |
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1162 |
\] |
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1163 |
We will establish the analogous isomorphism for a topological $A_\infty$ 1-cat $\cC$ |
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1164 |
and modules $\cM_\cC$ and $_\cC\cN$, |
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|
1165 |
\[ |
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|
1166 |
(\cM_\cC\ot {_\cC\cN})^* \cong \hom_\cC(\cM_\cC \to (_\cC\cN)^*) . |
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1167 |
\] |
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1168 |
|
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|
1169 |
In the next few paragraphs we define the objects appearing in the above equation: |
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1170 |
$\cM_\cC\ot {_\cC\cN}$, $(\cM_\cC\ot {_\cC\cN})^*$, $(_\cC\cN)^*$ and finally |
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1171 |
$\hom_\cC$. |
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1172 |
|
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1173 |
|
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1174 |
\def\olD{{\overline D}} |
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1175 |
\def\cbar{{\bar c}} |
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1176 |
In the previous subsection we defined a tensor product of $A_\infty$ $n$-category modules |
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1177 |
for general $n$. |
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1178 |
For $n=1$ this definition is a homotopy colimit indexed by subdivisions of a fixed interval $J$ |
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1179 |
and their gluings (antirefinements). |
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1180 |
(This tensor product depends functorially on the choice of $J$.) |
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1181 |
To a subdivision $D$ |
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1182 |
\[ |
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1183 |
J = I_1\cup \cdots\cup I_p |
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1184 |
\] |
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1185 |
we associate the chain complex |
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1186 |
\[ |
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1187 |
\psi(D) = \cM(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{m-1})\ot\cN(I_m) . |
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1188 |
\] |
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1189 |
To each antirefinement we associate a chain map using the composition law of $\cC$ and the |
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1190 |
module actions of $\cC$ on $\cM$ and $\cN$. |
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1191 |
The underlying graded vector space of the homotopy colimit is |
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1192 |
\[ |
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1193 |
\bigoplus_l \bigoplus_{\olD} \psi(D_0)[l] , |
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1194 |
\] |
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1195 |
where $l$ runs through the natural numbers, $\olD = (D_0\to D_1\to\cdots\to D_l)$ |
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1196 |
runs through chains of antirefinements of length $l+1$, and $[l]$ denotes a grading shift. |
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1197 |
We will denote an element of the summand indexed by $\olD$ by |
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1198 |
$\olD\ot m\ot\cbar\ot n$, where $m\ot\cbar\ot n \in \psi(D_0)$. |
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1199 |
The boundary map is given by |
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1200 |
\begin{align*} |
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1201 |
\bd(\olD\ot m\ot\cbar\ot n) &= (\bd_0 \olD)\ot \rho(m\ot\cbar\ot n) + (\bd_+ \olD)\ot m\ot\cbar\ot n \; + \\ |
291 | 1202 |
& \qquad + (-1)^l \olD\ot\bd m\ot\cbar\ot n + (-1)^{l+\deg m} \olD\ot m\ot\bd \cbar\ot n + \\ |
1203 |
& \qquad + (-1)^{l+\deg m + \deg \cbar} \olD\ot m\ot \cbar\ot \bd n |
|
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1204 |
\end{align*} |
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1205 |
where $\bd_+ \olD = \sum_{i>0} (-1)^i (D_0\to \cdots \to \widehat{D_i} \to \cdots \to D_l)$ (those parts of the simplicial |
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|
1206 |
boundary which retain $D_0$), $\bd_0 \olD = (D_1 \to \cdots \to D_l)$, |
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1207 |
and $\rho$ is the gluing map associated to the antirefinement $D_0\to D_1$. |
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1208 |
|
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1209 |
$(\cM_\cC\ot {_\cC\cN})^*$ is just the dual chain complex to $\cM_\cC\ot {_\cC\cN}$: |
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1210 |
\[ |
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1211 |
\prod_l \prod_{\olD} (\psi(D_0)[l])^* , |
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|
1212 |
\] |
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1213 |
where $(\psi(D_0)[l])^*$ denotes the linear dual. |
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1214 |
The boundary is given by |
291 | 1215 |
\begin{align} |
1216 |
\label{eq:tensor-product-boundary} |
|
1217 |
(-1)^{\deg f +1} (\bd f)(\olD\ot m\ot\cbar\ot n) & = f((\bd_0 \olD)\ot \rho(m\ot\cbar\ot n)) + f((\bd_+ \olD)\ot m\ot\cbar\ot n) + \\ |
|
1218 |
& \qquad + (-1)^{l} f(\olD\ot\bd m\ot\cbar \ot n) + (-1)^{l + \deg m} f(\olD\ot m\ot\bd \cbar \ot n) + \notag \\ |
|
1219 |
& \qquad + (-1)^{l + \deg m + \deg \cbar} f(\olD\ot m\ot\cbar\ot \bd n). \notag |
|
1220 |
\end{align} |
|
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1221 |
|
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1222 |
Next we define the dual module $(_\cC\cN)^*$. |
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1223 |
This will depend on a choice of interval $J$, just as the tensor product did. |
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1224 |
Recall that $_\cC\cN$ is, among other things, a functor from right-marked intervals |
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|
1225 |
to chain complexes. |
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1226 |
Given $J$, we define for each $K\sub J$ which contains the {\it left} endpoint of $J$ |
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|
1227 |
\[ |
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1228 |
(_\cC\cN)^*(K) \deq ({_\cC\cN}(J\setmin K))^* , |
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|
1229 |
\] |
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1230 |
where $({_\cC\cN}(J\setmin K))^*$ denotes the (linear) dual of the chain complex associated |
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1231 |
to the right-marked interval $J\setmin K$. |
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|
1232 |
This extends to a functor from all left-marked intervals (not just those contained in $J$). |
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|
1233 |
\nn{need to say more here; not obvious how homeomorphisms act} |
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|
1234 |
It's easy to verify the remaining module axioms. |
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1235 |
|
260 | 1236 |
Now we reinterpret $(\cM_\cC\ot {_\cC\cN})^*$ |
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|
1237 |
as some sort of morphism $\cM_\cC \to (_\cC\cN)^*$. |
260 | 1238 |
Let $f\in (\cM_\cC\ot {_\cC\cN})^*$. |
261
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|
1239 |
Let $\olD = (D_0\cdots D_l)$ be a chain of subdivisions with $D_0 = [J = I_1\cup\cdots\cup I_m]$. |
291 | 1240 |
Recall that for any subdivision $J = I_1\cup\cdots\cup I_p$, $(_\cC\cN)^*(I_1\cup\cdots\cup I_{p-1}) = (_\cC\cN(I_p))^*$. |
261
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|
1241 |
Then for each such $\olD$ we have a degree $l$ map |
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|
1242 |
\begin{eqnarray*} |
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|
1243 |
\cM(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) &\to& (_\cC\cN)^*(I_1\cup\cdots\cup I_{p-1}) \\ |
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|
1244 |
m\ot \cbar &\mapsto& [n\mapsto f(\olD\ot m\ot \cbar\ot n)] |
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|
1245 |
\end{eqnarray*} |
260 | 1246 |
|
261
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|
1247 |
We are almost ready to give the definition of morphisms between arbitrary modules |
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|
1248 |
$\cX_\cC$ and $\cY_\cC$. |
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|
1249 |
Note that the rightmost interval $I_m$ does not appear above, except implicitly in $\olD$. |
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|
1250 |
To fix this, we define subdivisions as antirefinements of left-marked intervals. |
261
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|
1251 |
Subdivisions are just the obvious thing, but antirefinements are defined to mimic |
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|
1252 |
the above antirefinements of the fixed interval $J$, but with the rightmost subinterval $I_m$ always |
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|
1253 |
omitted. |
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|
1254 |
More specifically, $D\to D'$ is an antirefinement if $D'$ is obtained from $D$ by |
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|
1255 |
gluing subintervals together and/or omitting some of the rightmost subintervals. |
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|
1256 |
(See Figure \ref{fig:lmar}.) |
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|
1257 |
\begin{figure}[t]\begin{equation*} |
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|
1258 |
\mathfig{.6}{tempkw/left-marked-antirefinements} |
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|
1259 |
\end{equation*}\caption{Antirefinements of left-marked intervals}\label{fig:lmar}\end{figure} |
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|
1260 |
|
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|
1261 |
Now we define the chain complex $\hom_\cC(\cX_\cC \to \cY_\cC)$. |
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|
1262 |
The underlying vector space is |
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|
1263 |
\[ |
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|
1264 |
\prod_l \prod_{\olD} \hom[l]\left( |
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|
1265 |
\cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) \to |
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|
1266 |
\cY(I_1\cup\cdots\cup I_{p-1}) \rule{0pt}{1.1em}\right) , |
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|
1267 |
\] |
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|
1268 |
where, as usual $\olD = (D_0\cdots D_l)$ is a chain of antirefinements |
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|
1269 |
(but now of left-marked intervals) and $D_0$ is the subdivision $I_1\cup\cdots\cup I_{p-1}$. |
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|
1270 |
$\hom[l](- \to -)$ means graded linear maps of degree $l$. |
260 | 1271 |
|
261
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|
1272 |
\nn{small issue (pun intended): |
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1273 |
the above is a vector space only if the class of subdivisions is a set, e.g. only if |
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1274 |
all of our left-marked intervals are contained in some universal interval (like $J$ above). |
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1275 |
perhaps we should give another version of the definition in terms of natural transformations of functors.} |
260 | 1276 |
|
261
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|
1277 |
Abusing notation slightly, we will denote elements of the above space by $g$, with |
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1278 |
\[ |
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|
1279 |
\olD\ot x \ot \cbar \mapsto g(\olD\ot x \ot \cbar) \in \cY(I_1\cup\cdots\cup I_{p-1}) . |
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1280 |
\] |
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1281 |
For fixed $D_0$ and $D_1$, let $\cbar = \cbar'\ot\cbar''$, where $\cbar'$ corresponds to the subintervals of $D_0$ which map to $D_1$ and $\cbar''$ corresponds to the subintervals |
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|
1282 |
which are dropped off the right side. |
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|
1283 |
(Either $\cbar'$ or $\cbar''$ might be empty.) |
291 | 1284 |
\nn{surely $\cbar'$ can't be empy: we don't allow $D_1$ to be empty.} |
1285 |
Translating from the boundary map for $(\cM_\cC\ot {_\cC\cN})^*$ appearing in Equation \eqref{eq:tensor-product-boundary}, |
|
261
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|
1286 |
we have |
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|
1287 |
\begin{eqnarray*} |
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|
1288 |
(\bd g)(\olD\ot x \ot \cbar) &=& \bd(g(\olD\ot x \ot \cbar)) + g(\olD\ot\bd(x\ot\cbar)) + \\ |
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|
1289 |
& & \;\; g((\bd_+\olD)\ot x\ot\cbar) + \gl(g((\bd_0\olD)\ot x\ot\cbar')\ot\cbar'') . |
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|
1290 |
\end{eqnarray*} |
291 | 1291 |
\nn{put in signs, rearrange terms to match order in previous formulas} |
261
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1292 |
Here $\gl$ denotes the module action in $\cY_\cC$. |
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|
1293 |
This completes the definition of $\hom_\cC(\cX_\cC \to \cY_\cC)$. |
260 | 1294 |
|
261
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|
1295 |
Note that if $\bd g = 0$, then each |
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|
1296 |
\[ |
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|
1297 |
g(\olD\ot -) : \cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) \to \cY(I_1\cup\cdots\cup I_{p-1}) |
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|
1298 |
\] |
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|
1299 |
constitutes a null homotopy of |
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|
1300 |
$g((\bd \olD)\ot -)$ (where the $g((\bd_0 \olD)\ot -)$ part of $g((\bd \olD)\ot -)$ |
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1301 |
should be interpreted as above). |
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|
1302 |
|
262
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1303 |
Define a {\it naive morphism} |
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|
1304 |
\nn{should consider other names for this} |
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|
1305 |
of modules to be a collection of {\it chain} maps |
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|
1306 |
\[ |
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|
1307 |
h_K : \cX(K)\to \cY(K) |
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|
1308 |
\] |
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|
1309 |
for each left-marked interval $K$. |
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|
1310 |
These are required to commute with gluing; |
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|
1311 |
for each subdivision $K = I_1\cup\cdots\cup I_q$ the following diagram commutes: |
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|
1312 |
\[ \xymatrix{ |
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|
1313 |
\cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) \ar[r]^{h_{I_0}\ot \id} |
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|
1314 |
\ar[d]_{\gl} & \cY(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) |
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|
1315 |
\ar[d]^{\gl} \\ |
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|
1316 |
\cX(K) \ar[r]^{h_{K}} & \cY(K) |
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|
1317 |
} \] |
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|
1318 |
Given such an $h$ we can construct a non-naive morphism $g$, with $\bd g = 0$, as follows. |
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|
1319 |
Define $g(\olD\ot - ) = 0$ if the length/degree of $\olD$ is greater than 0. |
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|
1320 |
If $\olD$ consists of the single subdivision $K = I_0\cup\cdots\cup I_q$ then define |
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|
1321 |
\[ |
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|
1322 |
g(\olD\ot x\ot \cbar) \deq h_K(\gl(x\ot\cbar)) . |
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|
1323 |
\] |
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|
1324 |
Trivially, we have $(\bd g)(\olD\ot x \ot \cbar) = 0$ if $\deg(\olD) > 1$. |
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|
1325 |
If $\deg(\olD) = 1$, $(\bd g) = 0$ is equivalent to the fact that $h$ commutes with gluing. |
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|
1326 |
If $\deg(\olD) = 0$, $(\bd g) = 0$ is equivalent to the fact |
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|
1327 |
that each $h_K$ is a chain map. |
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|
1328 |
|
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|
1329 |
\medskip |
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|
1330 |
|
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|
1331 |
Given $_\cC\cZ$ and $g: \cX_\cC \to \cY_\cC$ with $\bd g = 0$ as above, we next define a chain map |
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|
1332 |
\[ |
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|
1333 |
g\ot\id : \cX_\cC \ot {}_\cC\cZ \to \cY_\cC \ot {}_\cC\cZ . |
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|
1334 |
\] |
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|
1335 |
\nn{this is fairly straightforward, but the details are messy enough that I'm inclined |
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|
1336 |
to postpone writing it up, in the hopes that I'll think of a better way to organize things.} |
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|
1337 |
|
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|
1338 |
|
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|
1339 |
|
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|
1340 |
|
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|
1341 |
\medskip |
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|
1342 |
|
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|
1343 |
|
261
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|
1344 |
\nn{do we need to say anything about composing morphisms of modules?} |
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|
1345 |
|
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|
1346 |
\nn{should we define functors between $n$-cats in a similar way?} |
260 | 1347 |
|
258
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|
1348 |
|
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|
1349 |
\nn{...} |
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|
1350 |
|
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|
1351 |
|
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|
1352 |
|
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|
1353 |
|
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|
1354 |
|
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|
1355 |
|
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|
1356 |
|
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|
1357 |
|
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|
1358 |
|
117
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|
1359 |
\subsection{The $n{+}1$-category of sphere modules} |
218 | 1360 |
\label{ssec:spherecat} |
117
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|
1361 |
|
205 | 1362 |
In this subsection we define an $n{+}1$-category $\cS$ of ``sphere modules" |
1363 |
whose objects correspond to $n$-categories. |
|
259
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|
1364 |
When $n=2$ |
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|
1365 |
this is a version of the familiar algebras-bimodules-intertwiners 2-category. |
222
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|
1366 |
(Terminology: It is clearly appropriate to call an $S^0$ module a bimodule, |
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|
1367 |
but this is much less true for higher dimensional spheres, |
155 | 1368 |
so we prefer the term ``sphere module" for the general case.) |
144 | 1369 |
|
205 | 1370 |
The $0$- through $n$-dimensional parts of $\cC$ are various sorts of modules, and we describe |
1371 |
these first. |
|
259
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|
1372 |
The $n{+}1$-dimensional part of $\cS$ consists of intertwiners |
205 | 1373 |
(of garden-variety $1$-category modules associated to decorated $n$-balls). |
1374 |
We will see below that in order for these $n{+}1$-morphisms to satisfy all of |
|
1375 |
the duality requirements of an $n{+}1$-category, we will have to assume |
|
1376 |
that our $n$-categories and modules have non-degenerate inner products. |
|
1377 |
(In other words, we need to assume some extra duality on the $n$-categories and modules.) |
|
1378 |
||
1379 |
\medskip |
|
1380 |
||
1381 |
Our first task is to define an $n$-category $m$-sphere module, for $0\le m \le n-1$. |
|
1382 |
These will be defined in terms of certain classes of marked balls, very similarly |
|
1383 |
to the definition of $n$-category modules above. |
|
1384 |
(This, in turn, is very similar to our definition of $n$-category.) |
|
1385 |
Because of this similarity, we only sketch the definitions below. |
|
1386 |
||
1387 |
We start with 0-sphere modules, which also could reasonably be called (categorified) bimodules. |
|
1388 |
(For $n=1$ they are precisely bimodules in the usual, uncategorified sense.) |
|
1389 |
Define a 0-marked $k$-ball $(X, M)$, $1\le k \le n$, to be a pair homeomorphic to the standard |
|
1390 |
$(B^k, B^{k-1})$, where $B^{k-1}$ is properly embedded in $B^k$. |
|
209 | 1391 |
See Figure \ref{feb21a}. |
205 | 1392 |
Another way to say this is that $(X, M)$ is homeomorphic to $B^{k-1}\times([-1,1], \{0\})$. |
1393 |
||
209 | 1394 |
\begin{figure}[!ht] |
1395 |
\begin{equation*} |
|
1396 |
\mathfig{.85}{tempkw/feb21a} |
|
1397 |
\end{equation*} |
|
1398 |
\caption{0-marked 1-ball and 0-marked 2-ball} |
|
1399 |
\label{feb21a} |
|
1400 |
\end{figure} |
|
1401 |
||
205 | 1402 |
0-marked balls can be cut into smaller balls in various ways. |
1403 |
These smaller balls could be 0-marked or plain. |
|
1404 |
We can also take the boundary of a 0-marked ball, which is 0-marked sphere. |
|
1405 |
||
1406 |
Fix $n$-categories $\cA$ and $\cB$. |
|
1407 |
These will label the two halves of a 0-marked $k$-ball. |
|
1408 |
The 0-sphere module we define next will depend on $\cA$ and $\cB$ |
|
1409 |
(it's an $\cA$-$\cB$ bimodule), but we will suppress that from the notation. |
|
1410 |
||
1411 |
An $n$-category 0-sphere module $\cM$ is a collection of functors $\cM_k$ from the category |
|
1412 |
of 0-marked $k$-balls, $1\le k \le n$, |
|
1413 |
(with the two halves labeled by $\cA$ and $\cB$) to the category of sets. |
|
1414 |
If $k=n$ these sets should be enriched to the extent $\cA$ and $\cB$ are. |
|
1415 |
Given a decomposition of a 0-marked $k$-ball $X$ into smaller balls $X_i$, we have |
|
1416 |
morphism sets $\cA_k(X_i)$ (if $X_i$ lies on the $\cA$-labeled side) |
|
1417 |
or $\cB_k(X_i)$ (if $X_i$ lies on the $\cB$-labeled side) |
|
1418 |
or $\cM_k(X_i)$ (if $X_i$ intersects the marking and is therefore a smaller 0-marked ball). |
|
1419 |
Corresponding to this decomposition we have an action and/or composition map |
|
1420 |
from the product of these various sets into $\cM(X)$. |
|
1421 |
||
1422 |
\medskip |
|
107 | 1423 |
|
222
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|
1424 |
Part of the structure of an $n$-category 0-sphere module is captured by saying it is |
206 | 1425 |
a collection $\cD^{ab}$ of $n{-}1$-categories, indexed by pairs $(a, b)$ of objects (0-morphisms) |
1426 |
of $\cA$ and $\cB$. |
|
1427 |
Let $J$ be some standard 0-marked 1-ball (i.e.\ an interval with a marked point in its interior). |
|
1428 |
Given a $j$-ball $X$, $0\le j\le n-1$, we define |
|
1429 |
\[ |
|
1430 |
\cD(X) \deq \cM(X\times J) . |
|
1431 |
\] |
|
1432 |
The product is pinched over the boundary of $J$. |
|
1433 |
$\cD$ breaks into ``blocks" according to the restrictions to the pinched points of $X\times J$ |
|
209 | 1434 |
(see Figure \ref{feb21b}). |
206 | 1435 |
These restrictions are 0-morphisms $(a, b)$ of $\cA$ and $\cB$. |
107 | 1436 |
|
209 | 1437 |
\begin{figure}[!ht] |
1438 |
\begin{equation*} |
|
1439 |
\mathfig{1}{tempkw/feb21b} |
|
1440 |
\end{equation*} |
|
1441 |
\caption{The pinched product $X\times J$} |
|
1442 |
\label{feb21b} |
|
1443 |
\end{figure} |
|
1444 |
||
206 | 1445 |
More generally, consider an interval with interior marked points, and with the complements |
1446 |
of these points labeled by $n$-categories $\cA_i$ ($0\le i\le l$) and the marked points labeled |
|
1447 |
by $\cA_i$-$\cA_{i+1}$ bimodules $\cM_i$. |
|
209 | 1448 |
(See Figure \ref{feb21c}.) |
206 | 1449 |
To this data we can apply to coend construction as in Subsection \ref{moddecss} above |
1450 |
to obtain an $\cA_0$-$\cA_l$ bimodule and, forgetfully, an $n{-}1$-category. |
|
1451 |
This amounts to a definition of taking tensor products of bimodules over $n$-categories. |
|
205 | 1452 |
|
209 | 1453 |
\begin{figure}[!ht] |
1454 |
\begin{equation*} |
|
1455 |
\mathfig{1}{tempkw/feb21c} |
|
1456 |
\end{equation*} |
|
1457 |
\caption{Marked and labeled 1-manifolds} |
|
1458 |
\label{feb21c} |
|
1459 |
\end{figure} |
|
1460 |
||
206 | 1461 |
We could also similarly mark and label a circle, obtaining an $n{-}1$-category |
1462 |
associated to the marked and labeled circle. |
|
209 | 1463 |
(See Figure \ref{feb21c}.) |
206 | 1464 |
If the circle is divided into two intervals, we can think of this $n{-}1$-category |
1465 |
as the 2-ended tensor product of the two bimodules associated to the two intervals. |
|
1466 |
||
1467 |
\medskip |
|
1468 |
||
1469 |
Next we define $n$-category 1-sphere modules. |
|
1470 |
These are just representations of (modules for) $n{-}1$-categories associated to marked and labeled |
|
1471 |
circles (1-spheres) which we just introduced. |
|
1472 |
||
1473 |
Equivalently, we can define 1-sphere modules in terms of 1-marked $k$-balls, $2\le k\le n$. |
|
1474 |
Fix a marked (and labeled) circle $S$. |
|
209 | 1475 |
Let $C(S)$ denote the cone of $S$, a marked 2-ball (Figure \ref{feb21d}). |
207 | 1476 |
\nn{I need to make up my mind whether marked things are always labeled too. |
1477 |
For the time being, let's say they are.} |
|
1478 |
A 1-marked $k$-ball is anything homeomorphic to $B^j \times C(S)$, $0\le j\le n-2$, |
|
1479 |
where $B^j$ is the standard $j$-ball. |
|
1480 |
1-marked $k$-balls can be decomposed in various ways into smaller balls, which are either |
|
1481 |
smaller 1-marked $k$-balls or the product of an unmarked ball with a marked interval. |
|
1482 |
We now proceed as in the above module definitions. |
|
1483 |
||
209 | 1484 |
\begin{figure}[!ht] |
1485 |
\begin{equation*} |
|
1486 |
\mathfig{.4}{tempkw/feb21d} |
|
1487 |
\end{equation*} |
|
1488 |
\caption{Cone on a marked circle} |
|
1489 |
\label{feb21d} |
|
1490 |
\end{figure} |
|
1491 |
||
207 | 1492 |
A $n$-category 1-sphere module is, among other things, an $n{-}2$-category $\cD$ with |
1493 |
\[ |
|
1494 |
\cD(X) \deq \cM(X\times C(S)) . |
|
1495 |
\] |
|
1496 |
The product is pinched over the boundary of $C(S)$. |
|
1497 |
$\cD$ breaks into ``blocks" according to the restriction to the |
|
1498 |
image of $\bd C(S) = S$ in $X\times C(S)$. |
|
1499 |
||
1500 |
More generally, consider a 2-manifold $Y$ |
|
1501 |
(e.g.\ 2-ball or 2-sphere) marked by an embedded 1-complex $K$. |
|
1502 |
The components of $Y\setminus K$ are labeled by $n$-categories, |
|
1503 |
the edges of $K$ are labeled by 0-sphere modules, |
|
1504 |
and the 0-cells of $K$ are labeled by 1-sphere modules. |
|
1505 |
We can now apply the coend construction and obtain an $n{-}2$-category. |
|
1506 |
If $Y$ has boundary then this $n{-}2$-category is a module for the $n{-}1$-manifold |
|
1507 |
associated to the (marked, labeled) boundary of $Y$. |
|
1508 |
In particular, if $\bd Y$ is a 1-sphere then we get a 1-sphere module as defined above. |
|
1509 |
||
1510 |
\medskip |
|
1511 |
||
1512 |
It should now be clear how to define $n$-category $m$-sphere modules for $0\le m \le n-1$. |
|
1513 |
For example, there is an $n{-}2$-category associated to a marked, labeled 2-sphere, |
|
208 | 1514 |
and a 2-sphere module is a representation of such an $n{-}2$-category. |
207 | 1515 |
|
1516 |
\medskip |
|
1517 |
||
1518 |
We can now define the $n$- or less dimensional part of our $n{+}1$-category $\cS$. |
|
1519 |
Choose some collection of $n$-categories, then choose some collections of bimodules for |
|
1520 |
these $n$-categories, then choose some collection of 1-sphere modules for the various |
|
1521 |
possible marked 1-spheres labeled by the $n$-categories and bimodules, and so on. |
|
1522 |
Let $L_i$ denote the collection of $i{-}1$-sphere modules we have chosen. |
|
1523 |
(For convenience, we declare a $(-1)$-sphere module to be an $n$-category.) |
|
1524 |
There is a wide range of possibilities. |
|
1525 |
$L_0$ could contain infinitely many $n$-categories or just one. |
|
1526 |
For each pair of $n$-categories in $L_0$, $L_1$ could contain no bimodules at all or |
|
1527 |
it could contain several. |
|
208 | 1528 |
The only requirement is that each $k$-sphere module be a module for a $k$-sphere $n{-}k$-category |
1529 |
constructed out of labels taken from $L_j$ for $j<k$. |
|
1530 |
||
1531 |
We now define $\cS(X)$, for $X$ of dimension at most $n$, to be the set of all |
|
1532 |
cell-complexes $K$ embedded in $X$, with the codimension-$j$ parts of $(X, K)$ labeled |
|
1533 |
by elements of $L_j$. |
|
1534 |
As described above, we can think of each decorated $k$-ball as defining a $k{-}1$-sphere module |
|
1535 |
for the $n{-}k{+}1$-category associated to its decorated boundary. |
|
1536 |
Thus the $k$-morphisms of $\cS$ (for $k\le n$) can be thought |
|
1537 |
of as $n$-category $k{-}1$-sphere modules |
|
1538 |
(generalizations of bimodules). |
|
1539 |
On the other hand, we can equally think of the $k$-morphisms as decorations on $k$-balls, |
|
1540 |
and from this (official) point of view it is clear that they satisfy all of the axioms of an |
|
1541 |
$n{+}1$-category. |
|
1542 |
(All of the axioms for the less-than-$n{+}1$-dimensional part of an $n{+}1$-category, that is.) |
|
1543 |
||
1544 |
\medskip |
|
1545 |
||
1546 |
Next we define the $n{+}1$-morphisms of $\cS$. |
|
1547 |
||
1548 |
||
1549 |
||
1550 |
||
1551 |
||
1552 |
||
207 | 1553 |
|
1554 |
\nn{...} |
|
101 | 1555 |
|
1556 |
\medskip |
|
1557 |
\hrule |
|
1558 |
\medskip |
|
1559 |
||
95 | 1560 |
\nn{to be continued...} |
101 | 1561 |
\medskip |
98 | 1562 |
|
1563 |
||
208 | 1564 |
|
1565 |
||
1566 |
||
1567 |
||
98 | 1568 |
Stuff that remains to be done (either below or in an appendix or in a separate section or in |
1569 |
a separate paper): |
|
1570 |
\begin{itemize} |
|
1571 |
\item spell out what difference (if any) Top vs PL vs Smooth makes |
|
207 | 1572 |
\item discuss Morita equivalence |
130 | 1573 |
\item morphisms of modules; show that it's adjoint to tensor product |
139 | 1574 |
(need to define dual module for this) |
1575 |
\item functors |
|
98 | 1576 |
\end{itemize} |
1577 |
||
204 | 1578 |
\bigskip |
1579 |
||
1580 |
\hrule |
|
134 | 1581 |
\nn{Some salvaged paragraphs that we might want to work back in:} |
204 | 1582 |
\bigskip |
98 | 1583 |
|
134 | 1584 |
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.) |
1585 |
||
1586 |
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 |
|
1587 |
\begin{align*} |
|
1588 |
\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)), |
|
1589 |
\end{align*} |
|
1590 |
where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism. |
|
1591 |
||
1592 |
\hrule |