author | Kevin Walker <kevin@canyon23.net> |
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%!TEX root = ../blob1.tex |
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\def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip} |
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\def\mmpar#1#2#3{\smallskip\noindent{\bf #1} (#2). {\it #3} \smallskip} |
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\section{$n$-categories and their modules} |
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\label{sec:ncats} |
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\subsection{Definition of $n$-categories} |
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\label{ss:n-cat-def} |
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Before proceeding, we need more appropriate definitions of $n$-categories, |
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$A_\infty$ $n$-categories, 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. |
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In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. |
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Generally, these sets are indexed by instances of a certain typical shape. |
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Some $n$-category definitions model $k$-morphisms on the standard 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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Thus we expect to associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic |
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to the standard $k$-ball. |
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By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
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standard $k$-ball. |
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We {\it do not} assume that it is equipped with a |
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preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below. |
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Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on |
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the boundary), we want a corresponding |
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bijection of sets $f:\cC(X)\to \cC(Y)$. |
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(This will imply ``strong duality", among other things.) Putting these together, we have |
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\begin{axiom}[Morphisms] |
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\label{axiom:morphisms} |
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For each $0 \le k \le n$, we have a functor $\cC_k$ from |
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the category of $k$-balls and |
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homeomorphisms to the category of sets and bijections. |
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\end{axiom} |
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(Note: We 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 $Y$ (c.f. \cite{MR2079378}). |
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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-morphisms 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. |
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For example, in a pivotal tensor category, there are natural isomorphisms $\Hom{}{A}{B \tensor C} \isoto \Hom{}{B^* \tensor A}{C}$, etc. |
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(sometimes called ``Frobenius reciprocity''), which canonically identify all the morphism spaces which have the same boundary. |
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We prefer 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 colimit |
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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{lem} |
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\label{lem:spheres} |
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For each $1 \le k \le n$, we have a functor $\cl{\cC}_{k-1}$ from |
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the category of $k{-}1$-spheres and |
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homeomorphisms to the category of sets and bijections. |
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\end{lem} |
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We postpone the proof \todo{} of this result until after we've actually given all the axioms. |
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Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, |
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along with the data described in the other Axioms at lower levels. |
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%In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point. |
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\begin{axiom}[Boundaries]\label{nca-boundary} |
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For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
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These maps, for various $X$, comprise a natural transformation of functors. |
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\end{axiom} |
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(Note that the first ``$\bd$" above is part of the data for the category, |
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while the second is the ordinary boundary of manifolds.) |
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Given $c\in\cl{\cC}(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$. |
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Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
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The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
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all $c\in \cl{\cC}(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category |
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(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. \cite{MR2079378}.) |
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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 lemma follows from the colimit construction used to define $\cl{\cC}_{k-1}$ |
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on spheres. |
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\begin{lem}[Boundary from domain and range] |
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\label{lem:domain-and-range} |
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Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$, |
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$B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}). |
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Let $\cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2)$ denote the fibered product of the |
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two maps $\bd: \cC(B_i)\to \cl{\cC}(E)$. |
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Then we have an injective map |
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\[ |
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\gl_E : \cC(B_1) \times_{\\cl{cC}(E)} \cC(B_2) \into \cl{\cC}(S) |
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\] |
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which is natural with respect to the actions of homeomorphisms. |
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(When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product |
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becomes a normal product.) |
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\end{lem} |
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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 $\cl{\cC}(S)_E$ denote the image of $\gl_E$. |
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We will refer to elements of $\\cl{cC}(S)_E$ 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 $\cC(X)_E = \bd^{-1}(\cl{\cC}(\bd X)_E)$. |
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We will call the projection $\cl{\cC}(S)_E \to \cC(B_i)$ |
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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 $a\in \cl{\cC}(S)_E$. |
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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. |
195 | 202 |
In particular, we have restriction maps $\cC(X)_E \to \cC(B_i)$ |
203 |
($i = 1, 2$, notation from previous paragraph). |
|
204 |
These restriction maps can be thought of as |
|
205 |
domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$. |
|
94 | 206 |
|
207 |
||
208 |
Next we consider composition of morphisms. |
|
209 |
For $n$-categories which lack strong duality, one usually considers |
|
210 |
$k$ different types of composition of $k$-morphisms, each associated to a different direction. |
|
211 |
(For example, vertical and horizontal composition of 2-morphisms.) |
|
212 |
In the presence of strong duality, these $k$ distinct compositions are subsumed into |
|
213 |
one general type of composition which can be in any ``direction". |
|
214 |
||
187 | 215 |
\begin{axiom}[Composition] |
216 |
Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
|
179 | 217 |
and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
103 | 218 |
Let $E = \bd Y$, which is a $k{-}2$-sphere. |
94 | 219 |
Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
220 |
We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$. |
|
221 |
Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. |
|
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We have a map |
94 | 223 |
\[ |
224 |
\gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E |
|
225 |
\] |
|
226 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
227 |
to the intersection of the boundaries of $B$ and $B_i$. |
|
228 |
If $k < n$ we require that $\gl_Y$ is injective. |
|
187 | 229 |
(For $k=n$, see below.) |
230 |
\end{axiom} |
|
94 | 231 |
|
179 | 232 |
\begin{figure}[!ht] |
222
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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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246 |
\caption{From two balls to one ball.}\label{blah5}\end{figure} |
179 | 247 |
|
195 | 248 |
\begin{axiom}[Strict associativity] \label{nca-assoc} |
187 | 249 |
The composition (gluing) maps above are strictly associative. |
250 |
\end{axiom} |
|
102 | 251 |
|
179 | 252 |
\begin{figure}[!ht] |
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253 |
$$\mathfig{.65}{ncat/strict-associativity}$$ |
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254 |
\caption{An example of strict associativity.}\label{blah6}\end{figure} |
179 | 255 |
|
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256 |
We'll use the notations $a\bullet b$ as well as $a \cup b$ for the glued together field $\gl_Y(a, b)$. |
110 | 257 |
In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ |
195 | 258 |
a restriction map (one of many types of map so called) and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$. |
259 |
%Compositions of boundary and restriction maps will also be called restriction maps. |
|
260 |
%For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a |
|
261 |
%restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$. |
|
110 | 262 |
|
192 | 263 |
We will write $\cC(B)_Y$ for the image of $\gl_Y$ in $\cC(B)$. |
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264 |
We will call elements of $\cC(B)_Y$ morphisms which are `splittable along $Y$' or `transverse to $Y$'. |
192 | 265 |
We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$. |
109 | 266 |
|
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|
267 |
More generally, let $\alpha$ be a subdivision of a ball $X$ into smaller balls. |
193 | 268 |
Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from |
269 |
the smaller balls to $X$. |
|
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270 |
We say that elements of $\cC(X)_\alpha$ are morphisms which are `splittable along $\alpha$'. |
193 | 271 |
In situations where the subdivision is notationally anonymous, we will write |
272 |
$\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to) |
|
273 |
the unnamed subdivision. |
|
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|
274 |
If $\beta$ is a subdivision of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cl{\cC}(\bd X)_\beta)$; |
193 | 275 |
this can also be denoted $\cC(X)\spl$ if the context contains an anonymous |
276 |
subdivision of $\bd X$ and no competing subdivision of $X$. |
|
192 | 277 |
|
278 |
The above two composition axioms are equivalent to the following one, |
|
102 | 279 |
which we state in slightly vague form. |
280 |
||
281 |
\xxpar{Multi-composition:} |
|
282 |
{Given any decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball |
|
283 |
into small $k$-balls, there is a |
|
284 |
map from an appropriate subset (like a fibered product) |
|
193 | 285 |
of $\cC(B_1)\spl\times\cdots\times\cC(B_m)\spl$ to $\cC(B)\spl$, |
95 | 286 |
and these various $m$-fold composition maps satisfy an |
179 | 287 |
operad-type strict associativity condition (Figure \ref{blah7}).} |
288 |
||
289 |
\begin{figure}[!ht] |
|
290 |
$$\mathfig{.8}{tempkw/blah7}$$ |
|
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291 |
\caption{Operad composition and associativity}\label{blah7}\end{figure} |
95 | 292 |
|
293 |
The next axiom is related to identity morphisms, though that might not be immediately obvious. |
|
294 |
||
343 | 295 |
\begin{axiom}[Product (identity) morphisms, preliminary version] |
296 |
For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$, |
|
297 |
usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. |
|
298 |
These maps must satisfy the following conditions. |
|
299 |
\begin{enumerate} |
|
300 |
\item |
|
301 |
If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram |
|
302 |
\[ \xymatrix{ |
|
303 |
X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\ |
|
304 |
X \ar[r]^{f} & X' |
|
305 |
} \] |
|
306 |
commutes, then we have |
|
307 |
\[ |
|
308 |
\tilde{f}(a\times D) = f(a)\times D' . |
|
309 |
\] |
|
310 |
\item |
|
311 |
Product morphisms are compatible with gluing (composition) in both factors: |
|
312 |
\[ |
|
313 |
(a'\times D)\bullet(a''\times D) = (a'\bullet a'')\times D |
|
314 |
\] |
|
315 |
and |
|
316 |
\[ |
|
317 |
(a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') . |
|
318 |
\] |
|
319 |
\item |
|
320 |
Product morphisms are associative: |
|
321 |
\[ |
|
322 |
(a\times D)\times D' = a\times (D\times D') . |
|
323 |
\] |
|
324 |
(Here we are implicitly using functoriality and the obvious homeomorphism |
|
325 |
$(X\times D)\times D' \to X\times(D\times D')$.) |
|
326 |
\item |
|
327 |
Product morphisms are compatible with restriction: |
|
328 |
\[ |
|
329 |
\res_{X\times E}(a\times D) = a\times E |
|
330 |
\] |
|
331 |
for $E\sub \bd D$ and $a\in \cC(X)$. |
|
332 |
\end{enumerate} |
|
333 |
\end{axiom} |
|
334 |
||
335 |
We will need to strengthen the above preliminary version of the axiom to allow |
|
336 |
for products which are ``pinched" in various ways along their boundary. |
|
337 |
(See Figure xxxx.) |
|
344 | 338 |
(The need for a strengthened version will become apparent in appendix \ref{sec:comparing-defs} |
339 |
where we construct a traditional category from a topological category.) |
|
343 | 340 |
Define a {\it pinched product} to be a map |
341 |
\[ |
|
342 |
\pi: E\to X |
|
343 |
\] |
|
344 | 344 |
such that $E$ is a $k{+}m$-ball, $X$ is a $k$-ball ($m\ge 1$), and $\pi$ is locally modeled |
343 | 345 |
on a standard iterated degeneracy map |
346 |
\[ |
|
344 | 347 |
d: \Delta^{k+m}\to\Delta^k . |
343 | 348 |
\] |
349 |
In other words, \nn{each point has a neighborhood blah blah...} |
|
350 |
(We thank Kevin Costello for suggesting this approach.) |
|
351 |
||
344 | 352 |
Note that for each interior point $x\in X$, $\pi\inv(x)$ is an $m$-ball, |
343 | 353 |
and for for each boundary point $x\in\bd X$, $\pi\inv(x)$ is a ball of dimension |
344 | 354 |
$l \le m$, with $l$ depending on $x$. |
343 | 355 |
|
356 |
It is easy to see that a composition of pinched products is again a pinched product. |
|
357 |
||
358 |
A {\it sub pinched product} is a sub-$m$-ball $E'\sub E$ such that the restriction |
|
359 |
$\pi:E'\to \pi(E')$ is again a pinched product. |
|
360 |
A {union} of pinched products is a decomposition $E = \cup_i E_i$ |
|
361 |
such that each $E_i\sub E$ is a sub pinched product. |
|
362 |
(See Figure xxxx.) |
|
363 |
||
364 |
The product axiom will give a map $\pi^*:\cC(X)\to \cC(E)$ for each pinched product |
|
365 |
$\pi:E\to X$. |
|
344 | 366 |
Morphisms in the image of $\pi^*$ will be called product morphisms. |
343 | 367 |
Before stating the axiom, we illustrate it in our two motivating examples of $n$-categories. |
368 |
In the case where $\cC(X) = \{f: X\to T\}$, we define $\pi^*(f) = f\circ\pi$. |
|
344 | 369 |
In the case where $\cC(X)$ is the set of all labeled embedded cell complexes $K$ in $X$, |
370 |
define $\pi^*(K) = \pi\inv(K)$, with each codimension $i$ cell $\pi\inv(c)$ labeled by the |
|
371 |
same (traditional) $i$-morphism as the corresponding codimension $i$ cell $c$. |
|
343 | 372 |
|
373 |
||
374 |
\addtocounter{axiom}{-1} |
|
187 | 375 |
\begin{axiom}[Product (identity) morphisms] |
344 | 376 |
For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
377 |
there is a map $\pi^*:\cC(X)\to \cC(E)$. |
|
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378 |
These maps must satisfy the following conditions. |
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379 |
\begin{enumerate} |
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380 |
\item |
344 | 381 |
If $\pi:E\to X$ and $\pi':E'\to X'$ are pinched products, and |
382 |
if $f:X\to X'$ and $\tilde{f}:E \to E'$ are maps such that the diagram |
|
95 | 383 |
\[ \xymatrix{ |
344 | 384 |
E \ar[r]^{\tilde{f}} \ar[d]_{\pi} & E' \ar[d]^{\pi'} \\ |
95 | 385 |
X \ar[r]^{f} & X' |
386 |
} \] |
|
109 | 387 |
commutes, then we have |
388 |
\[ |
|
344 | 389 |
\pi'^*\circ f = \tilde{f}\circ \pi^*. |
109 | 390 |
\] |
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391 |
\item |
344 | 392 |
Product morphisms are compatible with gluing (composition). |
393 |
Let $\pi:E\to X$, $\pi_1:E_1\to X_1$, and $\pi_2:E_2\to X_2$ |
|
394 |
be pinched products with $E = E_1\cup E_2$. |
|
395 |
Let $a\in \cC(X)$, and let $a_i$ denote the restriction of $a$ to $X_i\sub X$. |
|
396 |
Then |
|
109 | 397 |
\[ |
344 | 398 |
\pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) . |
109 | 399 |
\] |
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400 |
\item |
344 | 401 |
Product morphisms are associative. |
402 |
If $\pi:E\to X$ and $\rho:D\to E$ and pinched products then |
|
109 | 403 |
\[ |
344 | 404 |
\rho^*\circ\pi^* = (\pi\circ\rho)^* . |
109 | 405 |
\] |
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406 |
\item |
344 | 407 |
Product morphisms are compatible with restriction. |
408 |
If we have a commutative diagram |
|
409 |
\[ \xymatrix{ |
|
410 |
D \ar@{^(->}[r] \ar[d]_{\rho} & E \ar[d]^{\pi} \\ |
|
411 |
Y \ar@{^(->}[r] & X |
|
412 |
} \] |
|
413 |
such that $\rho$ and $\pi$ are pinched products, then |
|
110 | 414 |
\[ |
344 | 415 |
\res_D\circ\pi^* = \rho^*\circ\res_Y . |
110 | 416 |
\] |
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417 |
\end{enumerate} |
187 | 418 |
\end{axiom} |
95 | 419 |
|
343 | 420 |
|
421 |
\medskip |
|
128 | 422 |
|
95 | 423 |
All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |
424 |
The last axiom (below), concerning actions of |
|
425 |
homeomorphisms in the top dimension $n$, distinguishes the two cases. |
|
426 |
||
427 |
We start with the plain $n$-category case. |
|
428 |
||
267 | 429 |
\begin{axiom}[Isotopy invariance in dimension $n$]{\textup{\textbf{[preliminary]}}} |
187 | 430 |
Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
95 | 431 |
to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
187 | 432 |
Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$. |
267 | 433 |
\end{axiom} |
96 | 434 |
|
174 | 435 |
This axiom needs to be strengthened to force product morphisms to act as the identity. |
103 | 436 |
Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. |
96 | 437 |
Let $J$ be a 1-ball (interval). |
438 |
We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$. |
|
122 | 439 |
(Here we use the ``pinched" version of $Y\times J$. |
440 |
\nn{need notation for this}) |
|
96 | 441 |
We define a map |
442 |
\begin{eqnarray*} |
|
443 |
\psi_{Y,J}: \cC(X) &\to& \cC(X) \\ |
|
444 |
a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) . |
|
445 |
\end{eqnarray*} |
|
142 | 446 |
(See Figure \ref{glue-collar}.) |
189 | 447 |
\begin{figure}[!ht] |
448 |
\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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\node(\i-small) at (\i.east) [circle through={($(\i.east)+(\srad,0)$)}] {}; |
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\foreach \n in {1,2} { |
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\fill (intersection \n of \i-small and \i) node(\i-intersection-\n) {} circle (2pt); |
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} |
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\begin{scope}[decoration={brace,amplitude=10,aspect=0.5}] |
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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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488 |
\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 | 489 |
\end{equation*} |
490 |
||
491 |
\caption{Extended homeomorphism.}\label{glue-collar}\end{figure} |
|
174 | 492 |
We say that $\psi_{Y,J}$ is {\it extended isotopic} to the identity map. |
493 |
\nn{bad terminology; fix it later} |
|
494 |
\nn{also need to make clear that plain old isotopic to the identity implies |
|
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extended isotopic} |
|
97 | 496 |
\nn{maybe remark that in some examples (e.g.\ ones based on sub cell complexes) |
497 |
extended isotopies are also plain isotopies, so |
|
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no extension necessary} |
|
96 | 499 |
It can be thought of as the action of the inverse of |
500 |
a map which projects a collar neighborhood of $Y$ onto $Y$. |
|
501 |
||
502 |
The revised axiom is |
|
503 |
||
267 | 504 |
\addtocounter{axiom}{-1} |
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\begin{axiom}{\textup{\textbf{[topological version]}} Extended isotopy invariance in dimension $n$} |
187 | 506 |
\label{axiom:extended-isotopies} |
507 |
Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
|
174 | 508 |
to the identity on $\bd X$ and is extended isotopic (rel boundary) to the identity. |
187 | 509 |
Then $f$ acts trivially on $\cC(X)$. |
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\end{axiom} |
96 | 511 |
|
512 |
\nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
|
94 | 513 |
|
97 | 514 |
\smallskip |
515 |
||
516 |
For $A_\infty$ $n$-categories, we replace |
|
517 |
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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519 |
||
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\addtocounter{axiom}{-1} |
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521 |
\begin{axiom}{\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$} |
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522 |
For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
97 | 523 |
\[ |
524 |
C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
|
525 |
\] |
|
526 |
Here $C_*$ means singular chains and $\Homeo_\bd(X)$ is the space of homeomorphisms of $X$ |
|
527 |
which fix $\bd X$. |
|
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These action maps are required to be associative up to homotopy |
|
529 |
\nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
|
236 | 530 |
a diagram like the one in Proposition \ref{CHprop} commutes. |
97 | 531 |
\nn{repeat diagram here?} |
187 | 532 |
\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} |
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\end{axiom} |
97 | 534 |
|
535 |
We should strengthen the above axiom to apply to families of extended homeomorphisms. |
|
109 | 536 |
To do this we need to explain how extended homeomorphisms form a topological space. |
97 | 537 |
Roughly, the set of $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
538 |
and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
|
539 |
\nn{need to also say something about collaring homeomorphisms.} |
|
540 |
\nn{this paragraph needs work.} |
|
541 |
||
103 | 542 |
Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
543 |
into a plain $n$-category (enriched over graded groups). |
|
97 | 544 |
\nn{say more here?} |
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545 |
In a different direction, if we enrich over topological spaces instead of chain complexes, |
97 | 546 |
we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
547 |
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 | 549 |
type $A_\infty$ $n$-category. |
550 |
||
99 | 551 |
\medskip |
97 | 552 |
|
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553 |
The alert reader will have already noticed that our definition of a (plain) $n$-category |
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554 |
is extremely similar to our definition of a topological system of fields. |
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555 |
There are two essential differences. |
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556 |
First, for the $n$-category definition we restrict our attention to balls |
99 | 557 |
(and their boundaries), while for fields we consider all manifolds. |
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558 |
Second, in category definition we directly impose isotopy |
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559 |
invariance in dimension $n$, while in the fields definition we have do not expect isotopy invariance on fields |
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560 |
but instead remember a subspace of local relations which contain differences of isotopic fields. |
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561 |
(Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.) |
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562 |
Thus a system of fields and local relations $(\cF,\cU)$ determines an $n$-category $\cC_ {\cF,\cU}$ simply by restricting our attention to |
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563 |
balls and, at level $n$, quotienting out by the local relations: |
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564 |
\begin{align*} |
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565 |
\cC_{\cF,\cU}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / \cU(B) & \text{when $k=n$.}\end{cases} |
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566 |
\end{align*} |
142 | 567 |
This $n$-category can be thought of as the local part of the fields. |
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568 |
Conversely, given a topological $n$-category we can construct a system of fields via |
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569 |
a colimit construction; see \S \ref{ss:ncat_fields} below. |
99 | 570 |
|
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571 |
\subsection{Examples of $n$-categories} |
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572 |
\label{ss:ncat-examples} |
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|
101 | 574 |
|
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575 |
We now describe several classes of examples of $n$-categories satisfying our axioms. |
101 | 576 |
|
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577 |
\begin{example}[Maps to a space] |
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578 |
\rm |
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579 |
\label{ex:maps-to-a-space}% |
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580 |
Fix a `target space' $T$, any topological space. |
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581 |
We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows. |
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582 |
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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584 |
For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo |
196 | 585 |
homotopies fixed on $\bd X$. |
101 | 586 |
(Note that homotopy invariance implies isotopy invariance.) |
587 |
For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
|
588 |
be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. |
|
313 | 589 |
|
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590 |
Recall we described a system of fields and local relations based on maps to $T$ in Example \ref{ex:maps-to-a-space(fields)} above. |
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591 |
Constructing a system of fields from $\pi_{\leq n}(T)$ recovers that example. |
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592 |
\end{example} |
101 | 593 |
|
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594 |
\begin{example}[Maps to a space, with a fiber] |
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595 |
\rm |
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596 |
\label{ex:maps-to-a-space-with-a-fiber}% |
196 | 597 |
We can modify the example above, by fixing a |
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598 |
closed $m$-manifold $F$, and defining $\pi^{\times F}_{\leq n}(T)(X) = \Maps(X \times F \to T)$, |
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599 |
otherwise leaving the definition in Example \ref{ex:maps-to-a-space} unchanged. |
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600 |
Taking $F$ to be a point recovers the previous case. |
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601 |
\end{example} |
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602 |
|
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603 |
\begin{example}[Linearized, twisted, maps to a space] |
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604 |
\rm |
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605 |
\label{ex:linearized-maps-to-a-space}% |
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606 |
We can linearize Examples \ref{ex:maps-to-a-space} and \ref{ex:maps-to-a-space-with-a-fiber} as follows. |
101 | 607 |
Let $\alpha$ be an $(n{+}m{+}1)$-cocycle on $T$ with values in a ring $R$ |
191
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608 |
(have in mind the trivial cocycle). |
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609 |
For $X$ of dimension less than $n$ define $\pi^{\alpha, \times F}_{\leq n}(T)(X)$ as before, ignoring $\alpha$. |
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610 |
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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|
611 |
the $R$-module of finite linear combinations of continuous maps from $X\times F$ to $T$, |
101 | 612 |
modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy |
191
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613 |
$h: X\times F\times I \to T$, then $a = \alpha(h)b$. |
101 | 614 |
\nn{need to say something about fundamental classes, or choose $\alpha$ carefully} |
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615 |
\end{example} |
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|
616 |
|
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617 |
The next example is only intended to be illustrative, as we don't specify which definition of a `traditional $n$-category' we intend. |
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618 |
Further, most of these definitions don't even have an agreed-upon notion of `strong duality', which we assume here. |
191
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619 |
\begin{example}[Traditional $n$-categories] |
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620 |
\rm |
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621 |
\label{ex:traditional-n-categories} |
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622 |
Given a `traditional $n$-category with strong duality' $C$ |
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623 |
define $\cC(X)$, for $X$ a $k$-ball with $k < n$, |
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624 |
to be the set of all $C$-labeled sub cell complexes of $X$ (c.f. \S \ref{sec:fields}). |
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625 |
For $X$ an $n$-ball and $c\in \cl{\cC}(\bd X)$, define $\cC(X; c)$ to be finite linear |
191
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626 |
combinations of $C$-labeled sub cell complexes of $X$ |
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627 |
modulo the kernel of the evaluation map. |
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628 |
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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629 |
with each cell labelled by the $m$-th iterated identity morphism of the corresponding cell for $a$. |
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630 |
More generally, start with an $n{+}m$-category $C$ and a closed $m$-manifold $F$. |
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631 |
Define $\cC(X)$, for $\dim(X) < n$, |
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632 |
to be the set of all $C$-labeled sub cell complexes of $X\times F$. |
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633 |
Define $\cC(X; c)$, for $X$ an $n$-ball, |
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634 |
to be the dual Hilbert space $A(X\times F; c)$. |
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635 |
\nn{refer elsewhere for details?} |
313 | 636 |
|
637 |
||
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638 |
Recall we described a system of fields and local relations based on a `traditional $n$-category' |
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639 |
$C$ in Example \ref{ex:traditional-n-categories(fields)} above. |
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|
640 |
Constructing a system of fields from $\cC$ recovers that example. |
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641 |
\todo{Except that it doesn't: pasting diagrams v.s. string diagrams.} |
191
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|
642 |
\end{example} |
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643 |
|
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644 |
Finally, we describe a version of the bordism $n$-category suitable to our definitions. |
204 | 645 |
|
646 |
\nn{should also include example of ncats coming from TQFTs, or refer ahead to where we discuss that example} |
|
647 |
||
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648 |
\newcommand{\Bord}{\operatorname{Bord}} |
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649 |
\begin{example}[The bordism $n$-category, plain version] |
191
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650 |
\rm |
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651 |
\label{ex:bordism-category} |
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652 |
For a $k$-ball $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional |
191
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653 |
submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse |
196 | 654 |
to $\bd X$. |
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655 |
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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656 |
we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism |
196 | 657 |
$W \to W'$ which restricts to the identity on the boundary. |
191
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658 |
\end{example} |
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659 |
|
196 | 660 |
%\nn{the next example might be an unnecessary distraction. consider deleting it.} |
101 | 661 |
|
196 | 662 |
%\begin{example}[Variation on the above examples] |
663 |
%We could allow $F$ to have boundary and specify boundary conditions on $X\times \bd F$, |
|
664 |
%for example product boundary conditions or take the union over all boundary conditions. |
|
665 |
%%\nn{maybe should not emphasize this case, since it's ``better" in some sense |
|
666 |
%%to think of these guys as affording a representation |
|
667 |
%%of the $n{+}1$-category associated to $\bd F$.} |
|
668 |
%\end{example} |
|
101 | 669 |
|
670 |
||
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671 |
%We have two main examples of $A_\infty$ $n$-categories, coming from maps to a target space and from the blob complex. |
101 | 672 |
|
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673 |
\begin{example}[Chains of maps to a space] |
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674 |
\rm |
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675 |
\label{ex:chains-of-maps-to-a-space} |
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|
676 |
We can modify Example \ref{ex:maps-to-a-space} above to define the fundamental $A_\infty$ $n$-category $\pi^\infty_{\le n}(T)$ of a topological space $T$. |
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|
677 |
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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|
678 |
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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679 |
$$C_*(\Maps_c(X\times F \to T)),$$ where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
101 | 680 |
and $C_*$ denotes singular chains. |
211 | 681 |
\nn{maybe should also mention version where we enrich over spaces rather than chain complexes} |
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682 |
\end{example} |
101 | 683 |
|
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684 |
See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to |
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|
685 |
homotopy the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
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|
686 |
|
279 | 687 |
\begin{example}[Blob complexes of balls (with a fiber)] |
688 |
\rm |
|
689 |
\label{ex:blob-complexes-of-balls} |
|
291 | 690 |
Fix an $n-k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$. |
691 |
We will define an $A_\infty$ $k$-category $\cC$. |
|
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692 |
When $X$ is a $m$-ball, with $m<k$, define $\cC(X) = \cE(X\times F)$. |
291 | 693 |
When $X$ is an $k$-ball, |
279 | 694 |
define $\cC(X; c) = \bc^\cE_*(X\times F; c)$ |
695 |
where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
|
696 |
\end{example} |
|
101 | 697 |
|
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|
698 |
This example will be essential for Theorem \ref{product_thm} below, which allows us to compute the blob complex of a product. |
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|
699 |
Notice that with $F$ a point, the above example is a construction turning a topological |
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|
700 |
$n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$. |
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|
701 |
We think of this as providing a `free resolution' of the topological $n$-category. |
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|
702 |
\todo{Say more here!} |
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|
703 |
In fact, there is also a trivial, but mostly uninteresting, way to do this: |
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|
704 |
we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, |
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|
705 |
and take $\CD{B}$ to act trivially. |
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|
706 |
|
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|
707 |
Be careful that the `free resolution' of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. |
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|
708 |
It's easy to see that with $n=0$, the corresponding system of fields is just |
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|
709 |
linear combinations of connected components of $T$, and the local relations are trivial. |
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|
710 |
There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
191
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|
711 |
|
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|
712 |
\begin{example}[The bordism $n$-category, $A_\infty$ version] |
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|
713 |
\rm |
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|
714 |
\label{ex:bordism-category-ainf} |
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|
715 |
blah blah \nn{to do...} |
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|
716 |
\end{example} |
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|
717 |
|
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|
718 |
|
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|
719 |
\begin{example}[$E_n$ algebras] |
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|
720 |
\rm |
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|
721 |
\label{ex:e-n-alg} |
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|
722 |
Let $\cE\cB_n$ be the operad of smooth embeddings of $k$ (little) |
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|
723 |
copies of the standard $n$-ball $B^n$ into another (big) copy of $B^n$. |
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|
724 |
The operad $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad. |
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|
725 |
(By peeling the little balls, we see that both are homotopic to the space of $k$ framed points |
309
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|
726 |
in $B^n$.) |
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727 |
|
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728 |
Let $A$ be an $\cE\cB_n$-algebra. |
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729 |
We will define an $A_\infty$ $n$-category $\cC^A$. |
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730 |
\nn{...} |
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731 |
\end{example} |
95 | 732 |
|
108 | 733 |
|
734 |
||
735 |
||
736 |
||
737 |
||
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738 |
%\subsection{From $n$-categories to systems of fields} |
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739 |
\subsection{From balls to manifolds} |
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740 |
\label{ss:ncat_fields} \label{ss:ncat-coend} |
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741 |
In this section we describe how to extend an $n$-category $\cC$ as described above |
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742 |
(of either the plain or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$. |
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743 |
This extension is a certain colimit, and we've chosen the notation to remind you of this. |
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744 |
That is, we show that functors $\cC_k$ satisfying the axioms above have a canonical extension |
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745 |
from $k$-balls to arbitrary $k$-manifolds. |
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746 |
Recall that we've already anticipated this construction in the previous section, |
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747 |
inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls, |
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748 |
so that we can state the boundary axiom for $\cC$ on $k+1$-balls. |
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749 |
In the case of plain $n$-categories, this construction factors into a construction of a |
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750 |
system of fields and local relations, followed by the usual TQFT definition of a |
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751 |
vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}. |
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752 |
For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. |
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753 |
Recall that we can take a plain $n$-category $\cC$ and pass to the `free resolution', |
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754 |
an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls (recall Example \ref{ex:blob-complexes-of-balls} above). |
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755 |
We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant |
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756 |
for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the same as the original blob complex for $M$ with coefficients in $\cC$. |
108 | 757 |
|
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758 |
We will first define the `cell-decomposition' poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
340
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759 |
An $n$-category $\cC$ provides a functor from this poset to the category of sets, |
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760 |
and we will define $\cC(W)$ as a suitable colimit |
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761 |
(or homotopy colimit in the $A_\infty$ case) of this functor. |
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|
762 |
We'll later give a more explicit description of this colimit. |
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763 |
In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-manifolds with boundary data), |
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764 |
then the resulting colimit is also enriched, that is, the set associated to $W$ splits into subsets according to boundary data, and each of these subsets has the appropriate structure (e.g. a vector space or chain complex). |
108 | 765 |
|
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766 |
\begin{defn} |
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767 |
Say that a `permissible decomposition' of $W$ is a cell decomposition |
108 | 768 |
\[ |
769 |
W = \bigcup_a X_a , |
|
770 |
\] |
|
142 | 771 |
where each closed top-dimensional cell $X_a$ is an embedded $k$-ball. |
191
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772 |
|
108 | 773 |
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
191
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774 |
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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775 |
|
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776 |
The category $\cell(W)$ has objects the permissible decompositions of $W$, |
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777 |
and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
191
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|
778 |
See Figure \ref{partofJfig} for an example. |
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|
779 |
\end{defn} |
119 | 780 |
|
781 |
\begin{figure}[!ht] |
|
782 |
\begin{equation*} |
|
222
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783 |
\mathfig{.63}{ncat/zz2} |
119 | 784 |
\end{equation*} |
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|
785 |
\caption{A small part of $\cell(W)$} |
119 | 786 |
\label{partofJfig} |
787 |
\end{figure} |
|
788 |
||
108 | 789 |
|
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790 |
|
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|
791 |
An $n$-category $\cC$ determines |
329
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|
792 |
a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
108 | 793 |
(possibly with additional structure if $k=n$). |
197 | 794 |
Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
795 |
and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries |
|
796 |
are splittable along this decomposition. |
|
797 |
%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 | 798 |
|
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|
799 |
\begin{defn} |
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|
800 |
Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
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|
801 |
For a decomposition $x = \bigcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset |
191
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|
802 |
\begin{equation} |
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803 |
\label{eq:psi-C} |
197 | 804 |
\psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl |
191
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|
805 |
\end{equation} |
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|
806 |
where the restrictions to the various pieces of shared boundaries amongst the cells |
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807 |
$X_a$ 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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|
808 |
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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|
809 |
\end{defn} |
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|
810 |
|
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|
811 |
When the $n$-category $\cC$ is enriched in some symmetric monoidal category $(A,\boxtimes)$, and $W$ is a |
197 | 812 |
closed $n$-manifold, the functor $\psi_{\cC;W}$ has target $A$ and |
340
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|
813 |
we replace the cartesian product of sets appearing in Equation \eqref{eq:psi-C} with the monoidal product $\boxtimes$. |
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|
814 |
(Moreover, $\psi_{\cC;W}(x)$ might be a subobject, rather than a subset, of the product.) |
197 | 815 |
Similar things are true if $W$ is an $n$-manifold with non-empty boundary and we |
816 |
fix a field on $\bd W$ |
|
817 |
(i.e. fix an element of the colimit associated to $\bd W$). |
|
191
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818 |
|
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819 |
Finally, we construct $\cC(W)$ as the appropriate colimit of $\psi_{\cC;W}$. |
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820 |
|
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|
821 |
\begin{defn}[System of fields functor] |
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822 |
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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823 |
That is, for each decomposition $x$ there is a map |
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|
824 |
$\psi_{\cC;W}(x)\to \cC(W)$, these maps are compatible with the refinement maps |
108 | 825 |
above, and $\cC(W)$ is universal with respect to these properties. |
191
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|
826 |
\end{defn} |
112 | 827 |
|
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|
828 |
\begin{defn}[System of fields functor, $A_\infty$ case] |
340
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|
829 |
When $\cC$ is an $A_\infty$ $n$-category, $\cC(W)$ for $W$ a $k$-manifold with $k < n$ |
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830 |
is defined as above, as the colimit of $\psi_{\cC;W}$. |
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|
831 |
When $W$ is an $n$-manifold, the chain complex $\cC(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$. |
191
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832 |
\end{defn} |
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833 |
|
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|
834 |
We can specify boundary data $c \in \cC(\bdy W)$, and define functors $\psi_{\cC;W,c}$ |
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|
835 |
with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$. |
111 | 836 |
|
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|
837 |
We now give a more concrete description of the colimit in each case. |
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838 |
If $\cC$ is enriched over vector spaces, and $W$ is an $n$-manifold, |
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|
839 |
we can take the vector space $\cC(W,c)$ to be the direct sum over all permissible decompositions of $W$ |
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|
840 |
\begin{equation*} |
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|
841 |
\cC(W,c) = \left( \bigoplus_x \psi_{\cC;W,c}(x)\right) \big/ K |
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842 |
\end{equation*} |
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843 |
where $K$ is the vector space spanned by elements $a - g(a)$, with |
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844 |
$a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
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|
845 |
\to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
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846 |
|
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847 |
In the $A_\infty$ case, enriched over chain complexes, the concrete description of the homotopy colimit |
197 | 848 |
is more involved. |
142 | 849 |
%\nn{should probably rewrite this to be compatible with some standard reference} |
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850 |
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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|
851 |
Such sequences (for all $m$) form a simplicial set in $\cell(W)$. |
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|
852 |
Define $V$ as a vector space via |
112 | 853 |
\[ |
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|
854 |
V = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
112 | 855 |
\] |
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|
856 |
where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$. |
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|
857 |
(Our homological conventions are non-standard: if a complex $U$ is concentrated in degree $0$, |
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|
858 |
the complex $U[m]$ is concentrated in degree $m$.) |
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|
859 |
We endow $V$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ |
112 | 860 |
summands plus another term using the differential of the simplicial set of $m$-sequences. |
861 |
More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
|
862 |
summand of $V$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
|
863 |
\[ |
|
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|
864 |
\bd (a, \bar{x}) = (\bd a, \bar{x}) + (-1)^{\deg{a}} (g(a), d_0(\bar{x})) + (-1)^{\deg{a}} \sum_{j=1}^k (-1)^{j} (a, d_j(\bar{x})) , |
112 | 865 |
\] |
866 |
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 | 867 |
is the usual gluing map coming from the antirefinement $x_0 \le x_1$. |
112 | 868 |
\nn{need to say this better} |
869 |
\nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which |
|
870 |
combine only two balls at a time; for $n=1$ this version will lead to usual definition |
|
871 |
of $A_\infty$ category} |
|
108 | 872 |
|
113 | 873 |
We will call $m$ the filtration degree of the complex. |
874 |
We can think of this construction as starting with a disjoint copy of a complex for each |
|
875 |
permissible decomposition (filtration degree 0). |
|
876 |
Then we glue these together with mapping cylinders coming from gluing maps |
|
877 |
(filtration degree 1). |
|
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878 |
Then we kill the extra homology we just introduced with mapping |
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|
879 |
cylinders between the mapping cylinders (filtration degree 2), and so on. |
113 | 880 |
|
108 | 881 |
$\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
882 |
||
883 |
It is easy to see that |
|
884 |
there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps |
|
885 |
comprise a natural transformation of functors. |
|
886 |
||
887 |
\nn{need to finish explaining why we have a system of fields; |
|
888 |
need to say more about ``homological" fields? |
|
889 |
(actions of homeomorphisms); |
|
890 |
define $k$-cat $\cC(\cdot\times W)$} |
|
891 |
||
892 |
\subsection{Modules} |
|
95 | 893 |
|
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|
894 |
Next we define plain and $A_\infty$ $n$-category modules. |
199 | 895 |
The definition will be very similar to that of $n$-categories, |
896 |
but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
|
109 | 897 |
\nn{** need to make sure all revisions of $n$-cat def are also made to module def.} |
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|
898 |
\nn{in particular, need to to get rid of the ``hemisphere axiom"} |
198 | 899 |
%\nn{should they be called $n$-modules instead of just modules? probably not, but worth considering.} |
900 |
||
104 | 901 |
Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
102 | 902 |
in the context of an $m{+}1$-dimensional TQFT. |
903 |
Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
|
904 |
This will be explained in more detail as we present the axioms. |
|
905 |
||
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|
906 |
\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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|
907 |
|
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|
908 |
Throughout, we fix an $n$-category $\cC$. |
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|
909 |
For all but one axiom, it doesn't matter whether $\cC$ is a topological $n$-category or an $A_\infty$ $n$-category. |
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|
910 |
We state the final axiom, on actions of homeomorphisms, differently in the two cases. |
102 | 911 |
|
912 |
Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair |
|
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|
913 |
$$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$ |
102 | 914 |
We call $B$ the ball and $N$ the marking. |
915 |
A homeomorphism between marked $k$-balls is a homeomorphism of balls which |
|
916 |
restricts to a homeomorphism of markings. |
|
917 |
||
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|
918 |
\begin{module-axiom}[Module morphisms] |
102 | 919 |
{For each $0 \le k \le n$, we have a functor $\cM_k$ from |
920 |
the category of marked $k$-balls and |
|
921 |
homeomorphisms to the category of sets and bijections.} |
|
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|
922 |
\end{module-axiom} |
102 | 923 |
|
924 |
(As with $n$-categories, we will usually omit the subscript $k$.) |
|
925 |
||
104 | 926 |
For example, let $\cD$ be the $m{+}1$-dimensional TQFT which assigns to a $k$-manifold $N$ the set |
927 |
of maps from $N$ to $T$, modulo homotopy (and possibly linearized) if $k=m$. |
|
928 |
Let $W$ be an $(m{-}n{+}1)$-dimensional manifold with boundary. |
|
929 |
Let $\cC$ be the $n$-category with $\cC(X) \deq \cD(X\times \bd W)$. |
|
930 |
Let $\cM(B, N) \deq \cD((B\times \bd W)\cup (N\times W))$. |
|
931 |
(The union is along $N\times \bd W$.) |
|
110 | 932 |
(If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be |
933 |
the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.) |
|
102 | 934 |
|
182 | 935 |
\begin{figure}[!ht] |
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|
936 |
$$\mathfig{.8}{ncat/boundary-collar}$$ |
182 | 937 |
\caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure} |
938 |
||
103 | 939 |
Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
940 |
Call such a thing a {marked $k{-}1$-hemisphere}. |
|
102 | 941 |
|
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|
942 |
\begin{lem} |
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|
943 |
\label{lem:hemispheres} |
102 | 944 |
{For each $0 \le k \le n-1$, we have a functor $\cM_k$ from |
104 | 945 |
the category of marked $k$-hemispheres and |
102 | 946 |
homeomorphisms to the category of sets and bijections.} |
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|
947 |
\end{lem} |
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|
948 |
The proof is exactly analogous to that of Lemma \ref{lem:spheres}, and we omit the details. |
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|
949 |
We use the same type of colimit construction. |
102 | 950 |
|
104 | 951 |
In our example, let $\cM(H) \deq \cD(H\times\bd W \cup \bd H\times W)$. |
952 |
||
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|
953 |
\begin{module-axiom}[Module boundaries (maps)] |
102 | 954 |
{For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cM(\bd M)$. |
955 |
These maps, for various $M$, comprise a natural transformation of functors.} |
|
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|
956 |
\end{module-axiom} |
102 | 957 |
|
110 | 958 |
Given $c\in\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
102 | 959 |
|
960 |
If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
|
961 |
then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$ |
|
962 |
and $c\in \cC(\bd M)$. |
|
963 |
||
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|
964 |
\begin{lem}[Boundary from domain and range] |
102 | 965 |
{Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k$-hemisphere ($0\le k\le n-1$), |
104 | 966 |
$M_i$ is a marked $k$-ball, and $E = M_1\cap M_2$ is a marked $k{-}1$-hemisphere. |
967 |
Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the |
|
968 |
two maps $\bd: \cM(M_i)\to \cM(E)$. |
|
102 | 969 |
Then (axiom) we have an injective map |
970 |
\[ |
|
199 | 971 |
\gl_E : \cM(M_1) \times_{\cM(E)} \cM(M_2) \hookrightarrow \cM(H) |
102 | 972 |
\] |
973 |
which is natural with respect to the actions of homeomorphisms.} |
|
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|
974 |
\end{lem} |
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|
975 |
Again, this is in exact analogy with Lemma \ref{lem:domain-and-range}. |
102 | 976 |
|
110 | 977 |
Let $\cM(H)_E$ denote the image of $\gl_E$. |
978 |
We will refer to elements of $\cM(H)_E$ as ``splittable along $E$" or ``transverse to $E$". |
|
979 |
||
980 |
||
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|
981 |
\begin{module-axiom}[Module to category restrictions] |
103 | 982 |
{For each marked $k$-hemisphere $H$ there is a restriction map |
983 |
$\cM(H)\to \cC(H)$. |
|
984 |
($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) |
|
985 |
These maps comprise a natural transformation of functors.} |
|
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|
986 |
\end{module-axiom} |
102 | 987 |
|
103 | 988 |
Note that combining the various boundary and restriction maps above |
110 | 989 |
(for both modules and $n$-categories) |
103 | 990 |
we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$ |
991 |
a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$. |
|
110 | 992 |
The subset is the subset of morphisms which are appropriately splittable (transverse to the |
993 |
cutting submanifolds). |
|
103 | 994 |
This fact will be used below. |
102 | 995 |
|
104 | 996 |
In our example, the various restriction and gluing maps above come from |
997 |
restricting and gluing maps into $T$. |
|
998 |
||
999 |
We require two sorts of composition (gluing) for modules, corresponding to two ways |
|
103 | 1000 |
of splitting a marked $k$-ball into two (marked or plain) $k$-balls. |
119 | 1001 |
(See Figure \ref{zzz3}.) |
103 | 1002 |
|
119 | 1003 |
\begin{figure}[!ht] |
1004 |
\begin{equation*} |
|
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|
1005 |
\mathfig{.4}{ncat/zz3} |
119 | 1006 |
\end{equation*} |
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|
1007 |
\caption{Module composition (top); $n$-category action (bottom).} |
119 | 1008 |
\label{zzz3} |
1009 |
\end{figure} |
|
1010 |
||
1011 |
First, we can compose two module morphisms to get another module morphism. |
|
103 | 1012 |
|
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|
1013 |
\begin{module-axiom}[Module composition] |
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|
1014 |
{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 | 1015 |
and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball. |
1016 |
Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere. |
|
1017 |
Note that each of $M$, $M_1$ and $M_2$ has its boundary split into two marked $k{-}1$-balls by $E$. |
|
1018 |
We have restriction (domain or range) maps $\cM(M_i)_E \to \cM(Y)$. |
|
1019 |
Let $\cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E$ denote the fibered product of these two maps. |
|
1020 |
Then (axiom) we have a map |
|
1021 |
\[ |
|
1022 |
\gl_Y : \cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E \to \cM(M)_E |
|
1023 |
\] |
|
1024 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
1025 |
to the intersection of the boundaries of $M$ and $M_i$. |
|
1026 |
If $k < n$ we require that $\gl_Y$ is injective. |
|
1027 |
(For $k=n$, see below.)} |
|
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|
1028 |
\end{module-axiom} |
119 | 1029 |
|
1030 |
||
103 | 1031 |
Second, we can compose an $n$-category morphism with a module morphism to get another |
1032 |
module morphism. |
|
1033 |
We'll call this the action map to distinguish it from the other kind of composition. |
|
1034 |
||
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|
1035 |
\begin{module-axiom}[$n$-category action] |
103 | 1036 |
{Let $M = X \cup_Y M'$, where $M$ and $M'$ are marked $k$-balls ($0\le k\le n$), |
1037 |
$X$ is a plain $k$-ball, |
|
1038 |
and $Y = X\cap M'$ is a $k{-}1$-ball. |
|
1039 |
Let $E = \bd Y$, which is a $k{-}2$-sphere. |
|
1040 |
We have restriction maps $\cM(M')_E \to \cC(Y)$ and $\cC(X)_E\to \cC(Y)$. |
|
1041 |
Let $\cC(X)_E \times_{\cC(Y)} \cM(M')_E$ denote the fibered product of these two maps. |
|
1042 |
Then (axiom) we have a map |
|
1043 |
\[ |
|
1044 |
\gl_Y :\cC(X)_E \times_{\cC(Y)} \cM(M')_E \to \cM(M)_E |
|
1045 |
\] |
|
1046 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
1047 |
to the intersection of the boundaries of $X$ and $M'$. |
|
1048 |
If $k < n$ we require that $\gl_Y$ is injective. |
|
1049 |
(For $k=n$, see below.)} |
|
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|
1050 |
\end{module-axiom} |
103 | 1051 |
|
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|
1052 |
\begin{module-axiom}[Strict associativity] |
103 | 1053 |
{The composition and action maps above are strictly associative.} |
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|
1054 |
\end{module-axiom} |
103 | 1055 |
|
110 | 1056 |
Note that the above associativity axiom applies to mixtures of module composition, |
1057 |
action maps and $n$-category composition. |
|
119 | 1058 |
See Figure \ref{zzz1b}. |
1059 |
||
1060 |
\begin{figure}[!ht] |
|
1061 |
\begin{equation*} |
|
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|
1062 |
\mathfig{0.49}{ncat/zz0} \mathfig{0.49}{ncat/zz1} |
119 | 1063 |
\end{equation*} |
1064 |
\caption{Two examples of mixed associativity} |
|
1065 |
\label{zzz1b} |
|
1066 |
\end{figure} |
|
1067 |
||
110 | 1068 |
|
1069 |
The above three axioms are equivalent to the following axiom, |
|
103 | 1070 |
which we state in slightly vague form. |
1071 |
\nn{need figure for this} |
|
1072 |
||
1073 |
\xxpar{Module multi-composition:} |
|
1074 |
{Given any decomposition |
|
1075 |
\[ |
|
1076 |
M = X_1 \cup\cdots\cup X_p \cup M_1\cup\cdots\cup M_q |
|
1077 |
\] |
|
1078 |
of a marked $k$-ball $M$ |
|
1079 |
into small (marked and plain) $k$-balls $M_i$ and $X_j$, there is a |
|
1080 |
map from an appropriate subset (like a fibered product) |
|
1081 |
of |
|
1082 |
\[ |
|
1083 |
\cC(X_1)\times\cdots\times\cC(X_p) \times \cM(M_1)\times\cdots\times\cM(M_q) |
|
1084 |
\] |
|
1085 |
to $\cM(M)$, |
|
1086 |
and these various multifold composition maps satisfy an |
|
1087 |
operad-type strict associativity condition.} |
|
1088 |
||
1089 |
(The above operad-like structure is analogous to the swiss cheese operad |
|
146 | 1090 |
\cite{MR1718089}.) |
200 | 1091 |
%\nn{need to double-check that this is true.} |
103 | 1092 |
|
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|
1093 |
\begin{module-axiom}[Product/identity morphisms] |
103 | 1094 |
{Let $M$ be a marked $k$-ball and $D$ be a plain $m$-ball, with $k+m \le n$. |
1095 |
Then we have a map $\cM(M)\to \cM(M\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cM(M)$. |
|
1096 |
If $f:M\to M'$ and $\tilde{f}:M\times D \to M'\times D'$ are maps such that the diagram |
|
1097 |
\[ \xymatrix{ |
|
1098 |
M\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & M'\times D' \ar[d]^{\pi} \\ |
|
1099 |
M \ar[r]^{f} & M' |
|
1100 |
} \] |
|
1101 |
commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.} |
|
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|
1102 |
\end{module-axiom} |
103 | 1103 |
|
111 | 1104 |
\nn{Need to add compatibility with various things, as in the n-cat version of this axiom above.} |
103 | 1105 |
|
200 | 1106 |
\nn{postpone finalizing the above axiom until the n-cat version is finalized} |
110 | 1107 |
|
103 | 1108 |
There are two alternatives for the next axiom, according whether we are defining |
1109 |
modules for plain $n$-categories or $A_\infty$ $n$-categories. |
|
1110 |
In the plain case we require |
|
1111 |
||
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|
1112 |
\begin{module-axiom}[\textup{\textbf{[topological version]}} Extended isotopy invariance in dimension $n$] |
103 | 1113 |
{Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts |
175 | 1114 |
to the identity on $\bd M$ and is extended isotopic (rel boundary) to the identity. |
103 | 1115 |
Then $f$ acts trivially on $\cM(M)$.} |
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|
1116 |
\end{module-axiom} |
103 | 1117 |
|
1118 |
\nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
|
1119 |
||
1120 |
We emphasize that the $\bd M$ above means boundary in the marked $k$-ball sense. |
|
1121 |
In other words, if $M = (B, N)$ then we require only that isotopies are fixed |
|
1122 |
on $\bd B \setmin N$. |
|
1123 |
||
1124 |
For $A_\infty$ modules we require |
|
1125 |
||
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|
1126 |
\addtocounter{module-axiom}{-1} |
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|
1127 |
\begin{module-axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act] |
103 | 1128 |
{For each marked $n$-ball $M$ and each $c\in \cM(\bd M)$ we have a map of chain complexes |
1129 |
\[ |
|
1130 |
C_*(\Homeo_\bd(M))\ot \cM(M; c) \to \cM(M; c) . |
|
1131 |
\] |
|
1132 |
Here $C_*$ means singular chains and $\Homeo_\bd(M)$ is the space of homeomorphisms of $M$ |
|
1133 |
which fix $\bd M$. |
|
1134 |
These action maps are required to be associative up to homotopy |
|
1135 |
\nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
|
236 | 1136 |
a diagram like the one in Proposition \ref{CHprop} commutes. |
103 | 1137 |
\nn{repeat diagram here?} |
1138 |
\nn{restate this with $\Homeo(M\to M')$? what about boundary fixing property?}} |
|
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|
1139 |
\end{module-axiom} |
103 | 1140 |
|
1141 |
\medskip |
|
102 | 1142 |
|
104 | 1143 |
Note that the above axioms imply that an $n$-category module has the structure |
1144 |
of an $n{-}1$-category. |
|
1145 |
More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$, |
|
1146 |
where $X$ is a $k$-ball or $k{-}1$-sphere and in the product $X\times J$ we pinch |
|
1147 |
above the non-marked boundary component of $J$. |
|
200 | 1148 |
(More specifically, we collapse $X\times P$ to a single point, where |
1149 |
$P$ is the non-marked boundary component of $J$.) |
|
1150 |
\nn{give figure for this?} |
|
104 | 1151 |
Then $\cE$ has the structure of an $n{-}1$-category. |
102 | 1152 |
|
105 | 1153 |
All marked $k$-balls are homeomorphic, unless $k = 1$ and our manifolds |
1154 |
are oriented or Spin (but not unoriented or $\text{Pin}_\pm$). |
|
1155 |
In this case ($k=1$ and oriented or Spin), there are two types |
|
1156 |
of marked 1-balls, call them left-marked and right-marked, |
|
1157 |
and hence there are two types of modules, call them right modules and left modules. |
|
1158 |
In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$), |
|
1159 |
there is no left/right module distinction. |
|
1160 |
||
130 | 1161 |
\medskip |
1162 |
||
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|
1163 |
We now give some examples of modules over topological and $A_\infty$ $n$-categories. |
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|
1164 |
|
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|
1165 |
\begin{example}[Examples from TQFTs] |
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|
1166 |
\todo{} |
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|
1167 |
\end{example} |
108 | 1168 |
|
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|
1169 |
\begin{example} |
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1170 |
Suppose $S$ is a topological space, with a subspace $T$. |
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|
1171 |
We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$ |
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|
1172 |
for $k<n$ the set $\pi_{\leq n}(S,T)(B,N)$ consists of all continuous maps of pairs |
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|
1173 |
$(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all |
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|
1174 |
such maps modulo homotopies fixed on $\bdy B \setminus N$. |
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|
1175 |
This is a module over the fundamental $n$-category $\pi_{\leq n}(S)$ of $S$, from Example \ref{ex:maps-to-a-space}. |
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|
1176 |
Modifications corresponding to Examples \ref{ex:maps-to-a-space-with-a-fiber} and |
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|
1177 |
\ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to |
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|
1178 |
Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains. |
224
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|
1179 |
\end{example} |
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|
1180 |
|
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|
1181 |
\subsection{Modules as boundary labels (colimits for decorated manifolds)} |
112 | 1182 |
\label{moddecss} |
108 | 1183 |
|
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1184 |
Fix a topological $n$-category or $A_\infty$ $n$-category $\cC$. |
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|
1185 |
Let $W$ be a $k$-manifold ($k\le n$), |
143 | 1186 |
let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$, |
1187 |
and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$. |
|
1188 |
||
1189 |
%Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), |
|
1190 |
%and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary |
|
1191 |
%component $\bd_i W$ of $W$. |
|
1192 |
%(More generally, each $\cN_i$ could label some codimension zero submanifold of $\bd W$.) |
|
108 | 1193 |
|
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|
1194 |
We will define a set $\cC(W, \cN)$ using a colimit construction similar to |
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|
1195 |
the one appearing in \S \ref{ss:ncat_fields} above. |
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|
1196 |
(If $k = n$ and our $n$-categories are enriched, then |
108 | 1197 |
$\cC(W, \cN)$ will have additional structure; see below.) |
1198 |
||
1199 |
Define a permissible decomposition of $W$ to be a decomposition |
|
1200 |
\[ |
|
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|
1201 |
W = \left(\bigcup_a X_a\right) \cup \left(\bigcup_{i,b} M_{ib}\right) , |
108 | 1202 |
\] |
1203 |
where each $X_a$ is a plain $k$-ball (disjoint from $\bd W$) and |
|
1204 |
each $M_{ib}$ is a marked $k$-ball intersecting $\bd_i W$, |
|
143 | 1205 |
with $M_{ib}\cap Y_i$ being the marking. |
1206 |
(See Figure \ref{mblabel}.) |
|
1207 |
\begin{figure}[!ht]\begin{equation*} |
|
286
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|
1208 |
\mathfig{.4}{ncat/mblabel} |
143 | 1209 |
\end{equation*}\caption{A permissible decomposition of a manifold |
340
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|
1210 |
whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$. |
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|
1211 |
Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel}\end{figure} |
108 | 1212 |
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
1213 |
of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
|
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|
1214 |
This defines a partial ordering $\cell(W)$, which we will think of as a category. |
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|
1215 |
(The objects of $\cell(D)$ are permissible decompositions of $W$, and there is a unique |
108 | 1216 |
morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
1217 |
||
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|
1218 |
The collection of modules $\cN$ determines |
329
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|
1219 |
a functor $\psi_\cN$ from $\cell(W)$ to the category of sets |
108 | 1220 |
(possibly with additional structure if $k=n$). |
329
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|
1221 |
For a decomposition $x = (X_a, M_{ib})$ in $\cell(W)$, define $\psi_\cN(x)$ to be the subset |
108 | 1222 |
\[ |
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|
1223 |
\psi_\cN(x) \sub \left(\prod_a \cC(X_a)\right) \times \left(\prod_{ib} \cN_i(M_{ib})\right) |
108 | 1224 |
\] |
1225 |
such that the restrictions to the various pieces of shared boundaries amongst the |
|
1226 |
$X_a$ and $M_{ib}$ all agree. |
|
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|
1227 |
(That is, the fibered product over the boundary maps.) |
108 | 1228 |
If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$ |
1229 |
via the gluing (composition or action) maps from $\cC$ and the $\cN_i$. |
|
1230 |
||
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|
1231 |
We now define the set $\cC(W, \cN)$ to be the colimit of the functor $\psi_\cN$. |
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|
1232 |
(As usual, if $k=n$ and we are in the $A_\infty$ case, then ``colimit" means |
143 | 1233 |
homotopy colimit.) |
108 | 1234 |
|
143 | 1235 |
If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define |
1236 |
$\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold |
|
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|
1237 |
$D\times Y_i \sub \bd(D\times W)$. |
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|
1238 |
It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$ |
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|
1239 |
has the structure of an $n{-}k$-category, which we call $\cT(W, \cN)(D)$. |
144 | 1240 |
|
1241 |
\medskip |
|
1242 |
||
1243 |
||
1244 |
We will use a simple special case of the above |
|
1245 |
construction to define tensor products |
|
1246 |
of modules. |
|
1247 |
Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$. |
|
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|
1248 |
(If $k=1$ and our manifolds are oriented, then one should be |
144 | 1249 |
a left module and the other a right module.) |
1250 |
Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$. |
|
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|
1251 |
Define the tensor product $\cM_1 \tensor \cM_2$ to be the |
340
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|
1252 |
$n{-}1$-category $\cT(J, \{\cM_1, \cM_2\})$. |
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|
1253 |
This of course depends (functorially) |
144 | 1254 |
on the choice of 1-ball $J$. |
105 | 1255 |
|
144 | 1256 |
We will define a more general self tensor product (categorified coend) below. |
1257 |
||
1258 |
%\nn{what about self tensor products /coends ?} |
|
105 | 1259 |
|
108 | 1260 |
\nn{maybe ``tensor product" is not the best name?} |
1261 |
||
144 | 1262 |
%\nn{start with (less general) tensor products; maybe change this later} |
106 | 1263 |
|
107 | 1264 |
|
1265 |
||
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|
1266 |
|
291 | 1267 |
\subsection{Morphisms of $A_\infty$ $1$-category modules} |
288 | 1268 |
\label{ss:module-morphisms} |
258
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1269 |
|
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|
1270 |
In order to state and prove our version of the higher dimensional Deligne conjecture |
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1271 |
(Section \ref{sec:deligne}), |
291 | 1272 |
we need to define morphisms of $A_\infty$ $1$-category modules and establish |
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1273 |
some of their elementary properties. |
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1274 |
|
340
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1275 |
To motivate the definitions which follow, consider algebras $A$ and $B$, |
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1276 |
right modules $X_B$ and $Z_A$ and a bimodule $\leftidx{_B}{Y}{_A}$, and the familiar adjunction |
258
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1277 |
\begin{eqnarray*} |
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|
1278 |
\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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1279 |
f &\mapsto& [x \mapsto f(x\ot -)] \\ |
279 | 1280 |
{}[x\ot y \mapsto g(x)(y)] & \mapsfrom & g . |
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1281 |
\end{eqnarray*} |
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1282 |
If $A$ and $Z_A$ are both the ground field $\k$, this simplifies to |
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1283 |
\[ |
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|
1284 |
(X_B\ot {_BY})^* \cong \hom_B(X_B \to (_BY)^*) . |
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1285 |
\] |
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|
1286 |
We will establish the analogous isomorphism for a topological $A_\infty$ 1-cat $\cC$ |
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1287 |
and modules $\cM_\cC$ and $_\cC\cN$, |
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|
1288 |
\[ |
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|
1289 |
(\cM_\cC\ot {_\cC\cN})^* \cong \hom_\cC(\cM_\cC \to (_\cC\cN)^*) . |
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|
1290 |
\] |
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|
1291 |
|
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|
1292 |
In the next few paragraphs we define the objects appearing in the above equation: |
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|
1293 |
$\cM_\cC\ot {_\cC\cN}$, $(\cM_\cC\ot {_\cC\cN})^*$, $(_\cC\cN)^*$ and finally |
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|
1294 |
$\hom_\cC$. |
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1295 |
|
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1296 |
|
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|
1297 |
\def\olD{{\overline D}} |
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|
1298 |
\def\cbar{{\bar c}} |
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|
1299 |
In the previous subsection we defined a tensor product of $A_\infty$ $n$-category modules |
258
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1300 |
for general $n$. |
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1301 |
For $n=1$ this definition is a homotopy colimit indexed by subdivisions of a fixed interval $J$ |
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1302 |
and their gluings (antirefinements). |
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|
1303 |
(This tensor product depends functorially on the choice of $J$.) |
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|
1304 |
To a subdivision $D$ |
258
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|
1305 |
\[ |
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|
1306 |
J = I_1\cup \cdots\cup I_p |
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1307 |
\] |
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|
1308 |
we associate the chain complex |
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|
1309 |
\[ |
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|
1310 |
\psi(D) = \cM(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{m-1})\ot\cN(I_m) . |
258
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|
1311 |
\] |
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|
1312 |
To each antirefinement we associate a chain map using the composition law of $\cC$ and the |
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|
1313 |
module actions of $\cC$ on $\cM$ and $\cN$. |
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|
1314 |
The underlying graded vector space of the homotopy colimit is |
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|
1315 |
\[ |
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|
1316 |
\bigoplus_l \bigoplus_{\olD} \psi(D_0)[l] , |
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|
1317 |
\] |
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|
1318 |
where $l$ runs through the natural numbers, $\olD = (D_0\to D_1\to\cdots\to D_l)$ |
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|
1319 |
runs through chains of antirefinements of length $l+1$, and $[l]$ denotes a grading shift. |
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|
1320 |
We will denote an element of the summand indexed by $\olD$ by |
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|
1321 |
$\olD\ot m\ot\cbar\ot n$, where $m\ot\cbar\ot n \in \psi(D_0)$. |
286
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|
1322 |
The boundary map is given by |
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|
1323 |
\begin{align*} |
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|
1324 |
\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 | 1325 |
& \qquad + (-1)^l \olD\ot\bd m\ot\cbar\ot n + (-1)^{l+\deg m} \olD\ot m\ot\bd \cbar\ot n + \\ |
1326 |
& \qquad + (-1)^{l+\deg m + \deg \cbar} \olD\ot m\ot \cbar\ot \bd n |
|
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|
1327 |
\end{align*} |
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|
1328 |
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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|
1329 |
boundary which retain $D_0$), $\bd_0 \olD = (D_1 \to \cdots \to D_l)$, |
259
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|
1330 |
and $\rho$ is the gluing map associated to the antirefinement $D_0\to D_1$. |
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|
1331 |
|
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|
1332 |
$(\cM_\cC\ot {_\cC\cN})^*$ is just the dual chain complex to $\cM_\cC\ot {_\cC\cN}$: |
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|
1333 |
\[ |
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|
1334 |
\prod_l \prod_{\olD} (\psi(D_0)[l])^* , |
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|
1335 |
\] |
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|
1336 |
where $(\psi(D_0)[l])^*$ denotes the linear dual. |
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|
1337 |
The boundary is given by |
291 | 1338 |
\begin{align} |
1339 |
\label{eq:tensor-product-boundary} |
|
1340 |
(-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) + \\ |
|
1341 |
& \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 \\ |
|
1342 |
& \qquad + (-1)^{l + \deg m + \deg \cbar} f(\olD\ot m\ot\cbar\ot \bd n). \notag |
|
1343 |
\end{align} |
|
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|
1344 |
|
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|
1345 |
Next we define the dual module $(_\cC\cN)^*$. |
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|
1346 |
This will depend on a choice of interval $J$, just as the tensor product did. |
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1347 |
Recall that $_\cC\cN$ is, among other things, a functor from right-marked intervals |
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|
1348 |
to chain complexes. |
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|
1349 |
Given $J$, we define for each $K\sub J$ which contains the {\it left} endpoint of $J$ |
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|
1350 |
\[ |
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|
1351 |
(_\cC\cN)^*(K) \deq ({_\cC\cN}(J\setmin K))^* , |
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|
1352 |
\] |
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|
1353 |
where $({_\cC\cN}(J\setmin K))^*$ denotes the (linear) dual of the chain complex associated |
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1354 |
to the right-marked interval $J\setmin K$. |
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|
1355 |
This extends to a functor from all left-marked intervals (not just those contained in $J$). |
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|
1356 |
\nn{need to say more here; not obvious how homeomorphisms act} |
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|
1357 |
It's easy to verify the remaining module axioms. |
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1358 |
|
260 | 1359 |
Now we reinterpret $(\cM_\cC\ot {_\cC\cN})^*$ |
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|
1360 |
as some sort of morphism $\cM_\cC \to (_\cC\cN)^*$. |
260 | 1361 |
Let $f\in (\cM_\cC\ot {_\cC\cN})^*$. |
261
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|
1362 |
Let $\olD = (D_0\cdots D_l)$ be a chain of subdivisions with $D_0 = [J = I_1\cup\cdots\cup I_m]$. |
291 | 1363 |
Recall that for any subdivision $J = I_1\cup\cdots\cup I_p$, $(_\cC\cN)^*(I_1\cup\cdots\cup I_{p-1}) = (_\cC\cN(I_p))^*$. |
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|
1364 |
Then for each such $\olD$ we have a degree $l$ map |
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|
1365 |
\begin{eqnarray*} |
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|
1366 |
\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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|
1367 |
m\ot \cbar &\mapsto& [n\mapsto f(\olD\ot m\ot \cbar\ot n)] |
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|
1368 |
\end{eqnarray*} |
260 | 1369 |
|
261
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|
1370 |
We are almost ready to give the definition of morphisms between arbitrary modules |
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1371 |
$\cX_\cC$ and $\cY_\cC$. |
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|
1372 |
Note that the rightmost interval $I_m$ does not appear above, except implicitly in $\olD$. |
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1373 |
To fix this, we define subdivisions as antirefinements of left-marked intervals. |
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1374 |
Subdivisions are just the obvious thing, but antirefinements are defined to mimic |
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|
1375 |
the above antirefinements of the fixed interval $J$, but with the rightmost subinterval $I_m$ always |
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|
1376 |
omitted. |
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|
1377 |
More specifically, $D\to D'$ is an antirefinement if $D'$ is obtained from $D$ by |
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|
1378 |
gluing subintervals together and/or omitting some of the rightmost subintervals. |
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|
1379 |
(See Figure \ref{fig:lmar}.) |
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|
1380 |
\begin{figure}[t]\begin{equation*} |
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|
1381 |
\mathfig{.6}{tempkw/left-marked-antirefinements} |
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|
1382 |
\end{equation*}\caption{Antirefinements of left-marked intervals}\label{fig:lmar}\end{figure} |
261
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1383 |
|
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1384 |
Now we define the chain complex $\hom_\cC(\cX_\cC \to \cY_\cC)$. |
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1385 |
The underlying vector space is |
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1386 |
\[ |
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1387 |
\prod_l \prod_{\olD} \hom[l]\left( |
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1388 |
\cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) \to |
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1389 |
\cY(I_1\cup\cdots\cup I_{p-1}) \rule{0pt}{1.1em}\right) , |
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|
1390 |
\] |
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1391 |
where, as usual $\olD = (D_0\cdots D_l)$ is a chain of antirefinements |
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1392 |
(but now of left-marked intervals) and $D_0$ is the subdivision $I_1\cup\cdots\cup I_{p-1}$. |
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1393 |
$\hom[l](- \to -)$ means graded linear maps of degree $l$. |
260 | 1394 |
|
261
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1395 |
\nn{small issue (pun intended): |
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1396 |
the above is a vector space only if the class of subdivisions is a set, e.g. only if |
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1397 |
all of our left-marked intervals are contained in some universal interval (like $J$ above). |
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1398 |
perhaps we should give another version of the definition in terms of natural transformations of functors.} |
260 | 1399 |
|
261
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1400 |
Abusing notation slightly, we will denote elements of the above space by $g$, with |
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1401 |
\[ |
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1402 |
\olD\ot x \ot \cbar \mapsto g(\olD\ot x \ot \cbar) \in \cY(I_1\cup\cdots\cup I_{p-1}) . |
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|
1403 |
\] |
340
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|
1404 |
For fixed $D_0$ and $D_1$, let $\cbar = \cbar'\ot\cbar''$, |
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|
1405 |
where $\cbar'$ corresponds to the subintervals of $D_0$ which map to $D_1$ and |
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|
1406 |
$\cbar''$ corresponds to the subintervals |
261
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1407 |
which are dropped off the right side. |
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|
1408 |
(Either $\cbar'$ or $\cbar''$ might be empty.) |
291 | 1409 |
\nn{surely $\cbar'$ can't be empy: we don't allow $D_1$ to be empty.} |
1410 |
Translating from the boundary map for $(\cM_\cC\ot {_\cC\cN})^*$ appearing in Equation \eqref{eq:tensor-product-boundary}, |
|
261
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|
1411 |
we have |
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|
1412 |
\begin{eqnarray*} |
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|
1413 |
(\bd g)(\olD\ot x \ot \cbar) &=& \bd(g(\olD\ot x \ot \cbar)) + g(\olD\ot\bd(x\ot\cbar)) + \\ |
330 | 1414 |
& & \;\; g((\bd_+\olD)\ot x\ot\cbar) + \gl''(g((\bd_0\olD)\ot \gl'(x\ot\cbar'))\ot\cbar'') . |
261
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|
1415 |
\end{eqnarray*} |
291 | 1416 |
\nn{put in signs, rearrange terms to match order in previous formulas} |
330 | 1417 |
Here $\gl''$ denotes the module action in $\cY_\cC$ |
1418 |
and $\gl'$ denotes the module action in $\cX_\cC$. |
|
261
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|
1419 |
This completes the definition of $\hom_\cC(\cX_\cC \to \cY_\cC)$. |
260 | 1420 |
|
261
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|
1421 |
Note that if $\bd g = 0$, then each |
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|
1422 |
\[ |
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|
1423 |
g(\olD\ot -) : \cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) \to \cY(I_1\cup\cdots\cup I_{p-1}) |
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|
1424 |
\] |
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|
1425 |
constitutes a null homotopy of |
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|
1426 |
$g((\bd \olD)\ot -)$ (where the $g((\bd_0 \olD)\ot -)$ part of $g((\bd \olD)\ot -)$ |
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|
1427 |
should be interpreted as above). |
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|
1428 |
|
262
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|
1429 |
Define a {\it naive morphism} |
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|
1430 |
\nn{should consider other names for this} |
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|
1431 |
of modules to be a collection of {\it chain} maps |
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|
1432 |
\[ |
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|
1433 |
h_K : \cX(K)\to \cY(K) |
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|
1434 |
\] |
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|
1435 |
for each left-marked interval $K$. |
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|
1436 |
These are required to commute with gluing; |
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|
1437 |
for each subdivision $K = I_1\cup\cdots\cup I_q$ the following diagram commutes: |
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|
1438 |
\[ \xymatrix{ |
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|
1439 |
\cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) \ar[r]^{h_{I_0}\ot \id} |
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|
1440 |
\ar[d]_{\gl} & \cY(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) |
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|
1441 |
\ar[d]^{\gl} \\ |
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|
1442 |
\cX(K) \ar[r]^{h_{K}} & \cY(K) |
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|
1443 |
} \] |
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|
1444 |
Given such an $h$ we can construct a non-naive morphism $g$, with $\bd g = 0$, as follows. |
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|
1445 |
Define $g(\olD\ot - ) = 0$ if the length/degree of $\olD$ is greater than 0. |
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|
1446 |
If $\olD$ consists of the single subdivision $K = I_0\cup\cdots\cup I_q$ then define |
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|
1447 |
\[ |
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|
1448 |
g(\olD\ot x\ot \cbar) \deq h_K(\gl(x\ot\cbar)) . |
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|
1449 |
\] |
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|
1450 |
Trivially, we have $(\bd g)(\olD\ot x \ot \cbar) = 0$ if $\deg(\olD) > 1$. |
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|
1451 |
If $\deg(\olD) = 1$, $(\bd g) = 0$ is equivalent to the fact that $h$ commutes with gluing. |
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|
1452 |
If $\deg(\olD) = 0$, $(\bd g) = 0$ is equivalent to the fact |
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|
1453 |
that each $h_K$ is a chain map. |
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|
1454 |
|
330 | 1455 |
We can think of a general closed element $g\in \hom_\cC(\cX_\cC \to \cY_\cC)$ |
1456 |
as a collection of chain maps which commute with the module action (gluing) up to coherent homotopy. |
|
1457 |
\nn{ideally should give explicit examples of this in low degrees, |
|
1458 |
but skip that for now.} |
|
1459 |
\nn{should also say something about composition of morphisms; well-defined up to homotopy, or maybe |
|
1460 |
should make some arbitrary choice} |
|
262
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|
1461 |
\medskip |
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|
1462 |
|
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|
1463 |
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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|
1464 |
\[ |
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|
1465 |
g\ot\id : \cX_\cC \ot {}_\cC\cZ \to \cY_\cC \ot {}_\cC\cZ . |
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|
1466 |
\] |
330 | 1467 |
|
1468 |
\nn{not sure whether to do low degree examples or try to state the general case; ideally both, |
|
1469 |
but maybe just low degrees for now.} |
|
1470 |
||
1471 |
||
1472 |
\nn{...} |
|
262
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|
1473 |
|
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|
1474 |
|
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|
1475 |
|
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|
1476 |
|
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|
1477 |
\medskip |
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|
1478 |
|
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|
1479 |
|
330 | 1480 |
\nn{should we define functors between $n$-cats in a similar way? i.e.\ natural transformations |
1481 |
of the $\cC$ functors which commute with gluing only up to higher morphisms? |
|
1482 |
perhaps worth having both definitions available. |
|
1483 |
certainly the simple kind (strictly commute with gluing) arise in nature.} |
|
258
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|
1484 |
|
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|
1485 |
|
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|
1486 |
|
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|
1487 |
|
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|
1488 |
|
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|
1489 |
|
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|
1490 |
|
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|
1491 |
|
117
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|
1492 |
\subsection{The $n{+}1$-category of sphere modules} |
218 | 1493 |
\label{ssec:spherecat} |
117
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|
1494 |
|
205 | 1495 |
In this subsection we define an $n{+}1$-category $\cS$ of ``sphere modules" |
327 | 1496 |
whose objects are $n$-categories. |
259
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|
1497 |
When $n=2$ |
327 | 1498 |
this is a version of the familiar algebras-bimodules-intertwiners $2$-category. |
1499 |
While it is clearly appropriate to call an $S^0$ module a bimodule, |
|
222
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|
1500 |
but this is much less true for higher dimensional spheres, |
327 | 1501 |
so we prefer the term ``sphere module" for the general case. |
144 | 1502 |
|
205 | 1503 |
The $0$- through $n$-dimensional parts of $\cC$ are various sorts of modules, and we describe |
1504 |
these first. |
|
259
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|
1505 |
The $n{+}1$-dimensional part of $\cS$ consists of intertwiners |
327 | 1506 |
of (garden-variety) $1$-category modules associated to decorated $n$-balls. |
205 | 1507 |
We will see below that in order for these $n{+}1$-morphisms to satisfy all of |
1508 |
the duality requirements of an $n{+}1$-category, we will have to assume |
|
1509 |
that our $n$-categories and modules have non-degenerate inner products. |
|
1510 |
(In other words, we need to assume some extra duality on the $n$-categories and modules.) |
|
1511 |
||
1512 |
\medskip |
|
1513 |
||
1514 |
Our first task is to define an $n$-category $m$-sphere module, for $0\le m \le n-1$. |
|
1515 |
These will be defined in terms of certain classes of marked balls, very similarly |
|
1516 |
to the definition of $n$-category modules above. |
|
1517 |
(This, in turn, is very similar to our definition of $n$-category.) |
|
1518 |
Because of this similarity, we only sketch the definitions below. |
|
1519 |
||
327 | 1520 |
We start with $0$-sphere modules, which also could reasonably be called (categorified) bimodules. |
205 | 1521 |
(For $n=1$ they are precisely bimodules in the usual, uncategorified sense.) |
327 | 1522 |
Define a $0$-marked $k$-ball $(X, M)$, $1\le k \le n$, to be a pair homeomorphic to the standard |
1523 |
$(B^k, B^{k-1})$. |
|
209 | 1524 |
See Figure \ref{feb21a}. |
205 | 1525 |
Another way to say this is that $(X, M)$ is homeomorphic to $B^{k-1}\times([-1,1], \{0\})$. |
1526 |
||
209 | 1527 |
\begin{figure}[!ht] |
1528 |
\begin{equation*} |
|
1529 |
\mathfig{.85}{tempkw/feb21a} |
|
1530 |
\end{equation*} |
|
1531 |
\caption{0-marked 1-ball and 0-marked 2-ball} |
|
1532 |
\label{feb21a} |
|
1533 |
\end{figure} |
|
1534 |
||
340
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1535 |
The $0$-marked balls can be cut into smaller balls in various ways. |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1536 |
We only consider those decompositions in which the smaller balls are either |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1537 |
$0$-marked (i.e. intersect the $0$-marking of the large ball in a disc) |
f7da004e1f14
breaking long lines (probably a waste of time, but I couldn't resist)
Kevin Walker <kevin@canyon23.net>
parents:
339
diff
changeset
|
1538 |
or plain (don't intersect the $0$-marking of the large ball). |
327 | 1539 |
We can also take the boundary of a $0$-marked ball, which is $0$-marked sphere. |
205 | 1540 |
|
1541 |
Fix $n$-categories $\cA$ and $\cB$. |
|
327 | 1542 |
These will label the two halves of a $0$-marked $k$-ball. |
1543 |
The $0$-sphere module we define next will depend on $\cA$ and $\cB$ |
|
205 | 1544 |
(it's an $\cA$-$\cB$ bimodule), but we will suppress that from the notation. |
1545 |
||
327 | 1546 |
An $n$-category $0$-sphere module $\cM$ is a collection of functors $\cM_k$ from the category |
1547 |
of $0$-marked $k$-balls, $1\le k \le n$, |
|
205 | 1548 |
(with the two halves labeled by $\cA$ and $\cB$) to the category of sets. |
1549 |
If $k=n$ these sets should be enriched to the extent $\cA$ and $\cB$ are. |
|
327 | 1550 |
Given a decomposition of a $0$-marked $k$-ball $X$ into smaller balls $X_i$, we have |
205 | 1551 |
morphism sets $\cA_k(X_i)$ (if $X_i$ lies on the $\cA$-labeled side) |
1552 |
or $\cB_k(X_i)$ (if $X_i$ lies on the $\cB$-labeled side) |
|
1553 |
or $\cM_k(X_i)$ (if $X_i$ intersects the marking and is therefore a smaller 0-marked ball). |
|
1554 |
Corresponding to this decomposition we have an action and/or composition map |
|
1555 |
from the product of these various sets into $\cM(X)$. |
|
1556 |
||
1557 |
\medskip |
|
107 | 1558 |
|
327 | 1559 |
Part of the structure of an $n$-category 0-sphere module $\cM$ is captured by saying it is |
206 | 1560 |
a collection $\cD^{ab}$ of $n{-}1$-categories, indexed by pairs $(a, b)$ of objects (0-morphisms) |
1561 |
of $\cA$ and $\cB$. |
|
1562 |
Let $J$ be some standard 0-marked 1-ball (i.e.\ an interval with a marked point in its interior). |
|
1563 |
Given a $j$-ball $X$, $0\le j\le n-1$, we define |
|
1564 |
\[ |
|
1565 |
\cD(X) \deq \cM(X\times J) . |
|
1566 |
\] |
|
1567 |
The product is pinched over the boundary of $J$. |
|
327 | 1568 |
The set $\cD$ breaks into ``blocks" according to the restrictions to the pinched points of $X\times J$ |
209 | 1569 |
(see Figure \ref{feb21b}). |
206 | 1570 |
These restrictions are 0-morphisms $(a, b)$ of $\cA$ and $\cB$. |
107 | 1571 |
|
209 | 1572 |
\begin{figure}[!ht] |
1573 |
\begin{equation*} |
|
1574 |
\mathfig{1}{tempkw/feb21b} |
|
1575 |
\end{equation*} |
|
1576 |
\caption{The pinched product $X\times J$} |
|
1577 |
\label{feb21b} |
|
1578 |
\end{figure} |
|
1579 |
||
206 | 1580 |
More generally, consider an interval with interior marked points, and with the complements |
1581 |
of these points labeled by $n$-categories $\cA_i$ ($0\le i\le l$) and the marked points labeled |
|
1582 |
by $\cA_i$-$\cA_{i+1}$ bimodules $\cM_i$. |
|
209 | 1583 |
(See Figure \ref{feb21c}.) |
327 | 1584 |
To this data we can apply the coend construction as in Subsection \ref{moddecss} above |
1585 |
to obtain an $\cA_0$-$\cA_l$ $0$-sphere module and, forgetfully, an $n{-}1$-category. |
|
1586 |
This amounts to a definition of taking tensor products of $0$-sphere module over $n$-categories. |
|
205 | 1587 |
|
209 | 1588 |
\begin{figure}[!ht] |
1589 |
\begin{equation*} |
|
1590 |
\mathfig{1}{tempkw/feb21c} |
|
1591 |
\end{equation*} |
|
1592 |
\caption{Marked and labeled 1-manifolds} |
|
1593 |
\label{feb21c} |
|
1594 |
\end{figure} |
|
1595 |
||
206 | 1596 |
We could also similarly mark and label a circle, obtaining an $n{-}1$-category |
1597 |
associated to the marked and labeled circle. |
|
209 | 1598 |
(See Figure \ref{feb21c}.) |
206 | 1599 |
If the circle is divided into two intervals, we can think of this $n{-}1$-category |
327 | 1600 |
as the 2-sided tensor product of the two bimodules associated to the two intervals. |
206 | 1601 |
|
1602 |
\medskip |
|
1603 |
||
1604 |
Next we define $n$-category 1-sphere modules. |
|
1605 |
These are just representations of (modules for) $n{-}1$-categories associated to marked and labeled |
|
1606 |
circles (1-spheres) which we just introduced. |
|
1607 |
||
1608 |
Equivalently, we can define 1-sphere modules in terms of 1-marked $k$-balls, $2\le k\le n$. |
|
1609 |
Fix a marked (and labeled) circle $S$. |
|
209 | 1610 |
Let $C(S)$ denote the cone of $S$, a marked 2-ball (Figure \ref{feb21d}). |
207 | 1611 |
\nn{I need to make up my mind whether marked things are always labeled too. |
1612 |
For the time being, let's say they are.} |
|
1613 |
A 1-marked $k$-ball is anything homeomorphic to $B^j \times C(S)$, $0\le j\le n-2$, |
|
1614 |
where $B^j$ is the standard $j$-ball. |
|
1615 |
1-marked $k$-balls can be decomposed in various ways into smaller balls, which are either |
|
1616 |
smaller 1-marked $k$-balls or the product of an unmarked ball with a marked interval. |
|
1617 |
We now proceed as in the above module definitions. |
|
1618 |
||
209 | 1619 |
\begin{figure}[!ht] |
1620 |
\begin{equation*} |
|
1621 |
\mathfig{.4}{tempkw/feb21d} |
|
1622 |
\end{equation*} |
|
1623 |
\caption{Cone on a marked circle} |
|
1624 |
\label{feb21d} |
|
1625 |
\end{figure} |
|
1626 |
||
207 | 1627 |
A $n$-category 1-sphere module is, among other things, an $n{-}2$-category $\cD$ with |
1628 |
\[ |
|
1629 |
\cD(X) \deq \cM(X\times C(S)) . |
|
1630 |
\] |
|
1631 |
The product is pinched over the boundary of $C(S)$. |
|
1632 |
$\cD$ breaks into ``blocks" according to the restriction to the |
|
1633 |
image of $\bd C(S) = S$ in $X\times C(S)$. |
|
1634 |
||
1635 |
More generally, consider a 2-manifold $Y$ |
|
1636 |
(e.g.\ 2-ball or 2-sphere) marked by an embedded 1-complex $K$. |
|
1637 |
The components of $Y\setminus K$ are labeled by $n$-categories, |
|
1638 |
the edges of $K$ are labeled by 0-sphere modules, |
|
1639 |
and the 0-cells of $K$ are labeled by 1-sphere modules. |
|
1640 |
We can now apply the coend construction and obtain an $n{-}2$-category. |
|
1641 |
If $Y$ has boundary then this $n{-}2$-category is a module for the $n{-}1$-manifold |
|
1642 |
associated to the (marked, labeled) boundary of $Y$. |
|
1643 |
In particular, if $\bd Y$ is a 1-sphere then we get a 1-sphere module as defined above. |
|
1644 |
||
1645 |
\medskip |
|
1646 |
||
1647 |
It should now be clear how to define $n$-category $m$-sphere modules for $0\le m \le n-1$. |
|
1648 |
For example, there is an $n{-}2$-category associated to a marked, labeled 2-sphere, |
|
208 | 1649 |
and a 2-sphere module is a representation of such an $n{-}2$-category. |
207 | 1650 |
|
1651 |
\medskip |
|
1652 |
||
1653 |
We can now define the $n$- or less dimensional part of our $n{+}1$-category $\cS$. |
|
1654 |
Choose some collection of $n$-categories, then choose some collections of bimodules for |
|
1655 |
these $n$-categories, then choose some collection of 1-sphere modules for the various |
|
1656 |
possible marked 1-spheres labeled by the $n$-categories and bimodules, and so on. |
|
1657 |
Let $L_i$ denote the collection of $i{-}1$-sphere modules we have chosen. |
|
1658 |
(For convenience, we declare a $(-1)$-sphere module to be an $n$-category.) |
|
1659 |
There is a wide range of possibilities. |
|
1660 |
$L_0$ could contain infinitely many $n$-categories or just one. |
|
1661 |
For each pair of $n$-categories in $L_0$, $L_1$ could contain no bimodules at all or |
|
1662 |
it could contain several. |
|
208 | 1663 |
The only requirement is that each $k$-sphere module be a module for a $k$-sphere $n{-}k$-category |
1664 |
constructed out of labels taken from $L_j$ for $j<k$. |
|
1665 |
||
1666 |
We now define $\cS(X)$, for $X$ of dimension at most $n$, to be the set of all |
|
1667 |
cell-complexes $K$ embedded in $X$, with the codimension-$j$ parts of $(X, K)$ labeled |
|
1668 |
by elements of $L_j$. |
|
1669 |
As described above, we can think of each decorated $k$-ball as defining a $k{-}1$-sphere module |
|
1670 |
for the $n{-}k{+}1$-category associated to its decorated boundary. |
|
1671 |
Thus the $k$-morphisms of $\cS$ (for $k\le n$) can be thought |
|
1672 |
of as $n$-category $k{-}1$-sphere modules |
|
1673 |
(generalizations of bimodules). |
|
1674 |
On the other hand, we can equally think of the $k$-morphisms as decorations on $k$-balls, |
|
1675 |
and from this (official) point of view it is clear that they satisfy all of the axioms of an |
|
1676 |
$n{+}1$-category. |
|
1677 |
(All of the axioms for the less-than-$n{+}1$-dimensional part of an $n{+}1$-category, that is.) |
|
1678 |
||
1679 |
\medskip |
|
1680 |
||
1681 |
Next we define the $n{+}1$-morphisms of $\cS$. |
|
1682 |
||
1683 |
||
1684 |
||
1685 |
||
1686 |
||
1687 |
||
207 | 1688 |
|
1689 |
\nn{...} |
|
101 | 1690 |
|
1691 |
\medskip |
|
1692 |
\hrule |
|
1693 |
\medskip |
|
1694 |
||
95 | 1695 |
\nn{to be continued...} |
101 | 1696 |
\medskip |
98 | 1697 |
|
1698 |
||
208 | 1699 |
|
1700 |
||
1701 |
||
1702 |
||
98 | 1703 |
Stuff that remains to be done (either below or in an appendix or in a separate section or in |
1704 |
a separate paper): |
|
1705 |
\begin{itemize} |
|
1706 |
\item spell out what difference (if any) Top vs PL vs Smooth makes |
|
207 | 1707 |
\item discuss Morita equivalence |
130 | 1708 |
\item morphisms of modules; show that it's adjoint to tensor product |
139 | 1709 |
(need to define dual module for this) |
1710 |
\item functors |
|
98 | 1711 |
\end{itemize} |
1712 |
||
204 | 1713 |