# HG changeset patch # User Scott Morrison # Date 1278992792 21600 # Node ID c4c1a01a900920983cb93635290b752b1402916b # Parent c5a35886cd8234c504e47ddc7b38c4ee82448ba9# Parent 35755232f6adcd38040cf0e5c319dd5f0a11a198 Automated merge with https://tqft.net/hg/blob/ diff -r 35755232f6ad -r c4c1a01a9009 blob1.tex --- a/blob1.tex Mon Jul 05 10:27:51 2010 -0700 +++ b/blob1.tex Mon Jul 12 21:46:32 2010 -0600 @@ -16,7 +16,7 @@ \maketitle -[revision $\ge$ 418; $\ge$ 5 July 2010] +[revision $\ge$ 427; $\ge$ 11 July 2010] {\color[rgb]{.9,.5,.2} \large \textbf{Draft version, read with caution.}} We're in the midst of revising this, and hope to have a version on the arXiv soon. @@ -33,12 +33,6 @@ check the argument about maps \item[9] (K) proofs trail off -\item Work in the references Chris Douglas gave us on the classification of local field theories, \cite{BDH-seminar,DSP-seminar,schommer-pries-thesis,0905.0465}. -\nn{KW: Do we need to do this? We don't really classify field theories. -I suppose our work could be interpreted as a alternative proof of cobordism hypothesis, but we -don't emphasize that at the moment. -On the other hand, I'm happy to do Chris a favor by citing this stuff.} - \item Make clear exactly what counts as a ``blob diagram", and search for ``blob diagram" diff -r 35755232f6ad -r c4c1a01a9009 text/a_inf_blob.tex --- a/text/a_inf_blob.tex Mon Jul 05 10:27:51 2010 -0700 +++ b/text/a_inf_blob.tex Mon Jul 12 21:46:32 2010 -0600 @@ -3,7 +3,7 @@ \section{The blob complex for $A_\infty$ $n$-categories} \label{sec:ainfblob} Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we make the anticlimactically tautological definition of the blob -complex $\bc_*(M;\cC)$ to be the homotopy colimit $\cl{\cC}(M)$ of Section \ref{ss:ncat_fields}. +complex $\bc_*(M;\cC)$ to be the homotopy colimit $\cl{\cC}(M)$ of \S\ref{ss:ncat_fields}. We will show below in Corollary \ref{cor:new-old} @@ -41,9 +41,9 @@ Given a topological $n$-category $C$ and a $n{-}k$-manifold $F$, recall from Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $\bc_*(F; C)$ defined by \begin{equation*} -\bc_*(F; C) = \cB_*(B \times F, C). +\bc_*(F; C)(B) = \cB_*(F \times B; C). \end{equation*} -Now, given a $k$-manifold $Y$, there is a homotopy equivalence between the `old-fashioned' +Now, given a $k$-manifold $Y$, there is a homotopy equivalence between the ``old-fashioned'' blob complex for $Y \times F$ with coefficients in $C$ and the ``new-fangled" (i.e.\ homotopy colimit) blob complex for $Y$ with coefficients in $\bc_*(F; C)$: \begin{align*} @@ -53,7 +53,7 @@ \begin{proof} -We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}. +We will use the concrete description of the colimit from \S\ref{ss:ncat_fields}. First we define a map \[ @@ -87,7 +87,7 @@ such that $a$ splits along $K_0\times F$ and $b$ is a generator appearing in an iterated boundary of $a$ (this includes $a$ itself). (Recall that $\ol{K} = (K_0,\ldots,K_l)$ denotes a chain of decompositions; -see Subsection \ref{ss:ncat_fields}.) +see \S\ref{ss:ncat_fields}.) By $(b, \ol{K})$ we really mean $(b^\sharp, \ol{K})$, where $b^\sharp$ is $b$ split according to $K_0\times F$. To simplify notation we will just write plain $b$ instead of $b^\sharp$. diff -r 35755232f6ad -r c4c1a01a9009 text/appendixes/comparing_defs.tex --- a/text/appendixes/comparing_defs.tex Mon Jul 05 10:27:51 2010 -0700 +++ b/text/appendixes/comparing_defs.tex Mon Jul 12 21:46:32 2010 -0600 @@ -3,7 +3,7 @@ \section{Comparing $n$-category definitions} \label{sec:comparing-defs} -In this appendix we relate the ``topological" category definitions of Section \ref{sec:ncats} +In this appendix we relate the ``topological" category definitions of \S\ref{sec:ncats} to more traditional definitions, for $n=1$ and 2. \nn{cases to cover: (a) plain $n$-cats for $n=1,2$; (b) $n$-cat modules for $n=1$, also 2?; diff -r 35755232f6ad -r c4c1a01a9009 text/basic_properties.tex --- a/text/basic_properties.tex Mon Jul 05 10:27:51 2010 -0700 +++ b/text/basic_properties.tex Mon Jul 12 21:46:32 2010 -0600 @@ -115,4 +115,4 @@ } This map is very far from being an isomorphism, even on homology. -We fix this deficit in Section \ref{sec:gluing} below. +We fix this deficit in \S\ref{sec:gluing} below. diff -r 35755232f6ad -r c4c1a01a9009 text/blobdef.tex --- a/text/blobdef.tex Mon Jul 05 10:27:51 2010 -0700 +++ b/text/blobdef.tex Mon Jul 12 21:46:32 2010 -0600 @@ -137,7 +137,8 @@ behavior} \nn{need to allow the case where $B\to X$ is not an embedding on $\bd B$. this is because any blob diagram on $X_{cut}$ should give rise to one on $X_{gl}$ -and blobs are allowed to meet $\bd X$.} +and blobs are allowed to meet $\bd X$. +Also, the complement of the blobs (and regions between nested blobs) might not be manifolds.} Now for the general case. A $k$-blob diagram consists of diff -r 35755232f6ad -r c4c1a01a9009 text/deligne.tex --- a/text/deligne.tex Mon Jul 05 10:27:51 2010 -0700 +++ b/text/deligne.tex Mon Jul 12 21:46:32 2010 -0600 @@ -44,7 +44,7 @@ We emphasize that in $\hom(\bc^C_*(I), \bc^C_*(I))$ we are thinking of $\bc^C_*(I)$ as a module for the $A_\infty$ 1-category associated to $\bd I$, and $\hom$ means the morphisms of such modules as defined in -Subsection \ref{ss:module-morphisms}. +\S\ref{ss:module-morphisms}. We can think of a fat graph as encoding a sequence of surgeries, starting at the bottommost interval of Figure \ref{delfig1} and ending at the topmost interval. @@ -215,7 +215,7 @@ \] which satisfy the operad compatibility conditions. On $C_0(FG^n_{\ol{M}\ol{N}})$ this agrees with the chain map $p$ defined above. -When $k=0$, this coincides with the $C_*(\Homeo(M_0\to N_0))$ action of Section \ref{sec:evaluation}. +When $k=0$, this coincides with the $C_*(\Homeo(M_0\to N_0))$ action of \S\ref{sec:evaluation}. \end{thm} If, in analogy to Hochschild cochains, we define elements of $\hom(M, N)$ diff -r 35755232f6ad -r c4c1a01a9009 text/evmap.tex --- a/text/evmap.tex Mon Jul 05 10:27:51 2010 -0700 +++ b/text/evmap.tex Mon Jul 12 21:46:32 2010 -0600 @@ -7,7 +7,7 @@ Let $CH_*(X, Y)$ denote $C_*(\Homeo(X \to Y))$, the singular chain complex of the space of homeomorphisms -between the $n$-manifolds $X$ and $Y$ (extending a fixed homeomorphism $\bd X \to \bd Y$). +between the $n$-manifolds $X$ and $Y$ (any given singular chain extends a fixed homeomorphism $\bd X \to \bd Y$). We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. (For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general than simplices --- they can be based on any linear polyhedron. @@ -24,12 +24,14 @@ $\Homeo(X, Y)$ on $\bc_*(X)$ (Property (\ref{property:functoriality})), and \item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, the following diagram commutes up to homotopy -\eq{ \xymatrix{ - CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]^(.7){e_{X\sgl Y\sgl}} \ar[d]^{\gl \otimes \gl} & \bc_*(Y\sgl) \ar[d]_{\gl} \\ +\begin{equation*} +\xymatrix@C+2cm{ CH_*(X, Y) \otimes \bc_*(X) - \ar@/_4ex/[r]_{e_{XY}} & - \bc_*(Y) -} } + \ar[r]_(.6){e_{XY}} \ar[d]^{\gl \otimes \gl} & + \bc_*(Y)\ar[d]^{\gl} \\ + CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]_(.6){e_{X\sgl Y\sgl}} & \bc_*(Y\sgl) +} +\end{equation*} \end{enumerate} Up to (iterated) homotopy, there is a unique family $\{e_{XY}\}$ of chain maps satisfying the above two conditions. @@ -69,13 +71,11 @@ Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. \end{lemma} -The proof will be given in Appendix \ref{sec:localising}. +The proof will be given in \S\ref{sec:localising}. \medskip Before diving into the details, we outline our strategy for the proof of Proposition \ref{CHprop}. - -%Suppose for the moment that evaluation maps with the advertised properties exist. Let $p$ be a singular cell in $CH_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$. We say that $p\ot b$ is {\it localizable} if there exists $V \sub X$ such that \begin{itemize} @@ -97,7 +97,7 @@ e_X(p\otimes b) = e_X(\gl(q\otimes b_V, r\otimes b_W)) = gl(e_{VV'}(q\otimes b_V), e_{WW'}(r\otimes b_W)) . \] -Since $r$ is a plain, 0-parameter family of homeomorphisms, we must have +Since $r$ is a 0-parameter family of homeomorphisms, we must have \[ e_{WW'}(r\otimes b_W) = r(b_W), \] @@ -122,7 +122,7 @@ Now for a little more detail. (But we're still just motivating the full, gory details, which will follow.) -Choose a metric on $X$, and let $\cU_\gamma$ be the open cover of by balls of radius $\gamma$. +Choose a metric on $X$, and let $\cU_\gamma$ be the open cover of balls of radius $\gamma$. By Lemma \ref{extension_lemma} we can restrict our attention to $k$-parameter families $p$ of homeomorphisms such that $\supp(p)$ is contained in the union of $k$ $\gamma$-balls. For fixed blob diagram $b$ and fixed $k$, it's not hard to show that for $\gamma$ small enough @@ -135,7 +135,7 @@ is the choice, for each localizable generator $p\ot b$, of disjoint balls $V$ containing $\supp(p)\cup\supp(b)$. Let $V'$ be another disjoint union of balls containing $\supp(p)\cup\supp(b)$, -and assume that there exists yet another disjoint union of balls $W$ with $W$ containing +and assume that there exists yet another disjoint union of balls $W$ containing $V\cup V'$. Then we can use $W$ to construct a homotopy between the two versions of $e_X$ associated to $V$ and $V'$. @@ -150,15 +150,15 @@ \medskip \begin{proof}[Proof of Proposition \ref{CHprop}.] -Notation: Let $|b| = \supp(b)$, $|p| = \supp(p)$. +We'll use the notation $|b| = \supp(b)$ and $|p| = \supp(p)$. Choose a metric on $X$. -Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging to zero +Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging monotonically to zero (e.g.\ $\ep_i = 2^{-i}$). Choose another sequence of positive real numbers $\delta_i$ such that $\delta_i/\ep_i$ converges monotonically to zero (e.g.\ $\delta_i = \ep_i^2$). Let $\phi_l$ be an increasing sequence of positive numbers -satisfying the inequalities of Lemma \ref{xx2phi}. +satisfying the inequalities of Lemma \ref{xx2phi} below. Given a generator $p\otimes b$ of $CH_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $l$ define \[ @@ -172,14 +172,14 @@ Next we define subcomplexes $G_*^{i,m} \sub CH_*(X)\otimes \bc_*(X)$. Let $p\ot b$ be a generator of $CH_*(X)\otimes \bc_*(X)$ and let $k = \deg(p\ot b) = \deg(p) + \deg(b)$. -$p\ot b$ is (by definition) in $G_*^{i,m}$ if either (a) $\deg(p) = 0$ or (b) +We say $p\ot b$ is in $G_*^{i,m}$ exactly when either (a) $\deg(p) = 0$ or (b) there exist codimension-zero submanifolds $V_0,\ldots,V_m \sub X$ such that each $V_j$ is homeomorphic to a disjoint union of balls and \[ N_{i,k}(p\ot b) \subeq V_0 \subeq N_{i,k+1}(p\ot b) \subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,k+m+1}(p\ot b) . \] -Further, we require (inductively) that $\bd(p\ot b) \in G_*^{i,m}$. +and further $\bd(p\ot b) \in G_*^{i,m}$. We also require that $b$ is splitable (transverse) along the boundary of each $V_l$. Note that $G_*^{i,m+1} \subeq G_*^{i,m}$. @@ -265,13 +265,12 @@ different choices of $V$ (and hence also different choices of $x'$) at each step. If $m \ge 1$ then $e$ and $\tilde{e}$ are homotopic. If $m \ge 2$ then any two choices of this first-order homotopy are second-order homotopic. -And so on. -In other words, $e : G_*^{i,m} \to \bc_*(X)$ is well-defined up to $m$-th order homotopy. +Continuing, $e : G_*^{i,m} \to \bc_*(X)$ is well-defined up to $m$-th order homotopy. \end{lemma} \begin{proof} We construct $h: G_*^{i,m} \to \bc_*(X)$ such that $\bd h + h\bd = e - \tilde{e}$. -$e$ and $\tilde{e}$ coincide on bidegrees $(0, j)$, so define $h$ +The chain maps $e$ and $\tilde{e}$ coincide on bidegrees $(0, j)$, so define $h$ to be zero there. Assume inductively that $h$ has been defined for degrees less than $k$. Let $p\ot b$ be a generator of degree $k$. @@ -344,11 +343,9 @@ \begin{proof} -Let $c$ be a subset of the blobs of $b$. -There exists $\lambda > 0$ such that $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ -and all such $c$. -(Here we are using a piecewise smoothness assumption for $\bd c$, and also -the fact that $\bd c$ is collared. + +There exists $\lambda > 0$ such that for every subset $c$ of the blobs of $b$ $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ . +(Here we are using the fact that the blobs are piecewise-linear and thatthat $\bd c$ is collared.) We need to consider all such $c$ because all generators appearing in iterated boundaries of $p\ot b$ must be in $G_*^{i,m}$.) @@ -414,7 +411,7 @@ is in $G_*^{i,m}$. \end{proof} -In the next few lemmas we have made no effort to optimize the various bounds. +In the next three lemmas, which provide the estimates needed above, we have made no effort to optimize the various bounds. (The bounds are, however, optimal in the sense of minimizing the amount of work we do. Equivalently, they are the first bounds we thought of.) @@ -431,7 +428,7 @@ Note that if $a \ge 2r$ then $\Nbd_a(S) \sup B_r(y)$. Let $z\in \Nbd_a(S) \setmin B_r(y)$. Consider the triangle -\nn{give figure?} with vertices $z$, $y$ and $s$ with $s\in S$. +with vertices $z$, $y$ and $s$ with $s\in S$. The length of the edge $yz$ is greater than $r$ which is greater than the length of the edge $ys$. It follows that the angle at $z$ is less than $\pi/2$ (less than $\pi/3$, in fact), @@ -442,7 +439,7 @@ If we replace $\ebb^n$ above with an arbitrary compact Riemannian manifold $M$, the same result holds, so long as $a$ is not too large: -\nn{what about PL? TOP?} +\nn{replace this with a PL version} \begin{lemma} \label{xxzz11} Let $M$ be a compact Riemannian manifold. @@ -498,7 +495,9 @@ where $t = \phi_{k-1}(2\phi_{k-1}+2)r + \phi_{k-1} r$. \end{proof} -\medskip + +We now return to defining the chain maps $e_X$. + Let $R_*$ be the chain complex with a generating 0-chain for each non-negative integer and a generating 1-chain connecting each adjacent pair $(j, j+1)$. @@ -592,13 +591,12 @@ the action maps $e_{X\sgl}$ and $e_X$. The gluing map $X\sgl\to X$ induces a map \[ - \gl: R_*\ot CH_*(X\sgl, X \sgl) \otimes \bc_*(X \sgl) \to R_*\ot CH_*(X, X) \otimes \bc_*(X) , + \gl: R_*\ot CH_*(X, X) \otimes \bc_*(X) \to R_*\ot CH_*(X\sgl, X \sgl) \otimes \bc_*(X \sgl) , \] and it is easy to see that $\gl(G^m_*)\sub \ol{G}^m_*$. From this it follows that the diagram in the statement of Proposition \ref{CHprop} commutes. -\medskip - +\todo{this paragraph isn't very convincing, or at least I don't see what's going on} Finally we show that the action maps defined above are independent of the choice of metric (up to iterated homotopy). The arguments are very similar to ones given above, so we only sketch them. @@ -614,6 +612,8 @@ We now observe that $CH_*(X, X) \ot \bc_*(X)$ retracts to $F_*$. Similar arguments show that this homotopy from $e$ to $e'$ is well-defined up to second order homotopy, and so on. + +This completes the proof of Proposition \ref{CHprop}. \end{proof} @@ -623,8 +623,8 @@ Let $b$ be a blob diagram and $p:P\times X\to X$ be a family of homeomorphisms. Then we may choose $e$ such that $e(p\ot b)$ is a sum of generators, each of which has support close to $p(t,|b|)$ for some $t\in P$. -More precisely, the support of the generators is contained in a small neighborhood -of $p(t,|b|)$ union some small balls. +More precisely, the support of the generators is contained in the union of a small neighborhood +of $p(t,|b|)$ with some small balls. (Here ``small" is in terms of the metric on $X$ that we chose to construct $e$.) \end{rem*} diff -r 35755232f6ad -r c4c1a01a9009 text/intro.tex --- a/text/intro.tex Mon Jul 05 10:27:51 2010 -0700 +++ b/text/intro.tex Mon Jul 12 21:46:32 2010 -0600 @@ -139,7 +139,7 @@ in order to better integrate it into the current intro.} As a starting point, consider TQFTs constructed via fields and local relations. -(See Section \ref{sec:tqftsviafields} or \cite{kw:tqft}.) +(See \S\ref{sec:tqftsviafields} or \cite{kw:tqft}.) This gives a satisfactory treatment for semisimple TQFTs (i.e.\ TQFTs for which the cylinder 1-category associated to an $n{-}1$-manifold $Y$ is semisimple for all $Y$). diff -r 35755232f6ad -r c4c1a01a9009 text/ncat.tex --- a/text/ncat.tex Mon Jul 05 10:27:51 2010 -0700 +++ b/text/ncat.tex Mon Jul 12 21:46:32 2010 -0600 @@ -97,7 +97,7 @@ $1\le k \le n$. At first it might seem that we need another axiom for this, but in fact once we have all the axioms in this subsection for $0$ through $k-1$ we can use a colimit -construction, as described in Subsection \ref{ss:ncat-coend} below, to extend $\cC_{k-1}$ +construction, as described in \S\ref{ss:ncat-coend} below, to extend $\cC_{k-1}$ to spheres (and any other manifolds): \begin{lem} @@ -127,6 +127,7 @@ The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and all $c\in \cl{\cC}(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category (e.g.\ vector spaces, or modules over some ring, or chain complexes), +\nn{actually, need both disj-union/sub and product/tensor-product; what's the name for this sort of cat?} and all the structure maps of the $n$-category should be compatible with the auxiliary category structure. Note that this auxiliary structure is only in dimension $n$; @@ -252,6 +253,8 @@ The composition (gluing) maps above are strictly associative. \end{axiom} +\nn{should say this means $N$ at a time, not just 3 at a time} + \begin{figure}[!ht] $$\mathfig{.65}{ncat/strict-associativity}$$ \caption{An example of strict associativity.}\label{blah6}\end{figure} @@ -378,7 +381,6 @@ \[ d: \Delta^{k+m}\to\Delta^k . \] -In other words, \nn{each point has a neighborhood blah blah...} (We thank Kevin Costello for suggesting this approach.) Note that for each interior point $x\in X$, $\pi\inv(x)$ is an $m$-ball, @@ -491,7 +493,7 @@ \] \item Product morphisms are associative. -If $\pi:E\to X$ and $\rho:D\to E$ and pinched products then +If $\pi:E\to X$ and $\rho:D\to E$ are pinched products then \[ \rho^*\circ\pi^* = (\pi\circ\rho)^* . \] @@ -518,7 +520,7 @@ We start with the plain $n$-category case. -\begin{axiom}[Isotopy invariance in dimension $n$]{\textup{\textbf{[preliminary]}}} +\begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$] Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$. @@ -592,7 +594,7 @@ The revised axiom is \addtocounter{axiom}{-1} -\begin{axiom}{\textup{\textbf{[topological version]}} Extended isotopy invariance in dimension $n$.} +\begin{axiom}[\textup{\textbf{[plain version]}} Extended isotopy invariance in dimension $n$.] \label{axiom:extended-isotopies} Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts to the identity on $\bd X$ and isotopic (rel boundary) to the identity. @@ -610,7 +612,7 @@ \addtocounter{axiom}{-1} -\begin{axiom}{\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.} +\begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes \[ C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . @@ -628,7 +630,7 @@ and we can replace the class of all intervals $J$ with intervals contained in $\r$. Having chains on the space of collar maps act gives rise to coherence maps involving weak identities. -We will not pursue this in this draft of the paper. +We will not pursue this in detail here. Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category into a plain $n$-category (enriched over graded groups). @@ -669,7 +671,7 @@ \begin{example}[Maps to a space] \rm \label{ex:maps-to-a-space}% -Let $T$be a topological space. +Let $T$ be a topological space. We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows. For $X$ a $k$-ball with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of all continuous maps from $X$ to $T$. @@ -687,7 +689,7 @@ an n-cat} } -\begin{example}[Maps to a space, with a fiber] +\begin{example}[Maps to a space, with a fiber] \label{ex:maps-with-fiber} \rm \label{ex:maps-to-a-space-with-a-fiber}% We can modify the example above, by fixing a @@ -711,8 +713,22 @@ Alternatively, we could equip the balls with fundamental classes.) \end{example} -The next example is only intended to be illustrative, as we don't specify which definition of a ``traditional $n$-category" we intend. -Further, most of these definitions don't even have an agreed-upon notion of ``strong duality", which we assume here. +\begin{example}[$n$-categories from TQFTs] +\rm +\label{ex:ncats-from-tqfts}% +Let $\cF$ be a TQFT in the sense of \S\ref{sec:fields}: an $n$-dimensional +system of fields (also denoted $\cF$) and local relations. +Let $W$ be an $n{-}j$-manifold. +Define the $j$-category $\cF(W)$ as follows. +If $X$ is a $k$-ball with $k}[r] \ar[d]_{\rho} & E \ar[d]^{\pi} \\ + Y \ar@{^(->}[r] & M +} \] +such that $\rho$ and $\pi$ are pinched products, then +\[ + \res_D\circ\pi^* = \rho^*\circ\res_Y . +\] +($Y$ could be either a marked or plain ball.) +\end{enumerate} \end{module-axiom} -\nn{Need to add compatibility with various things, as in the n-cat version of this axiom above.} +As in the $n$-category definition, once we have product morphisms we can define +collar maps $\cM(M)\to \cM(M)$. +Note that there are two cases: +the collar could intersect the marking of the marked ball $M$, in which case +we use a product on a morphism of $\cM$; or the collar could be disjoint from the marking, +in which case we use a product on a morphism of $\cC$. -\nn{postpone finalizing the above axiom until the n-cat version is finalized} +In our example, elements $a$ of $\cM(M)$ maps to $T$, and $\pi^*(a)$ is the pullback of +$a$ along a map associated to $\pi$. + +\medskip There are two alternatives for the next axiom, according whether we are defining modules for plain $n$-categories or $A_\infty$ $n$-categories. In the plain case we require -\begin{module-axiom}[\textup{\textbf{[topological version]}} Extended isotopy invariance in dimension $n$] +\begin{module-axiom}[\textup{\textbf{[plain version]}} Extended isotopy invariance in dimension $n$] {Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts -to the identity on $\bd M$ and is extended isotopic (rel boundary) to the identity. +to the identity on $\bd M$ and is isotopic (rel boundary) to the identity. Then $f$ acts trivially on $\cM(M)$.} +In addition, collar maps act trivially on $\cM(M)$. \end{module-axiom} -\nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} - We emphasize that the $\bd M$ above means boundary in the marked $k$-ball sense. In other words, if $M = (B, N)$ then we require only that isotopies are fixed on $\bd B \setmin N$. @@ -1290,19 +1365,19 @@ \addtocounter{module-axiom}{-1} \begin{module-axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act] -{For each marked $n$-ball $M$ and each $c\in \cM(\bd M)$ we have a map of chain complexes +For each marked $n$-ball $M$ and each $c\in \cM(\bd M)$ we have a map of chain complexes \[ C_*(\Homeo_\bd(M))\ot \cM(M; c) \to \cM(M; c) . \] Here $C_*$ means singular chains and $\Homeo_\bd(M)$ is the space of homeomorphisms of $M$ which fix $\bd M$. -These action maps are required to be associative up to homotopy -\nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that +These action maps are required to be associative up to homotopy, +and also compatible with composition (gluing) in the sense that a diagram like the one in Proposition \ref{CHprop} commutes. -\nn{repeat diagram here?} -\nn{restate this with $\Homeo(M\to M')$? what about boundary fixing property?}} \end{module-axiom} +As with the $n$-category version of the above axiom, we should also have families of collar maps act. + \medskip Note that the above axioms imply that an $n$-category module has the structure @@ -1312,7 +1387,6 @@ above the non-marked boundary component of $J$. (More specifically, we collapse $X\times P$ to a single point, where $P$ is the non-marked boundary component of $J$.) -\nn{give figure for this?} Then $\cE$ has the structure of an $n{-}1$-category. All marked $k$-balls are homeomorphic, unless $k = 1$ and our manifolds @@ -1328,20 +1402,29 @@ We now give some examples of modules over topological and $A_\infty$ $n$-categories. \begin{example}[Examples from TQFTs] -\todo{} +\rm +Continuing Example \ref{ex:ncats-from-tqfts}, with $\cF$ a TQFT, $W$ an $n{-}j$-manifold, +and $\cF(W)$ the $j$-category associated to $W$. +Let $Y$ be an $(n{-}j{+}1)$-manifold with $\bd Y = W$. +Define a $\cF(W)$ module $\cF(Y)$ as follows. +If $M = (B, N)$ is a marked $k$-ball with $k