--- 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"
--- 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$.
--- 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?;
--- 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.
--- 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
--- 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 $\hom(M, N)$
--- 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*}
--- 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 $Y$).
--- 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 $k<j$, let $\cF(W)(X) \deq \cF(W\times X)$.
+If $X$ is a $j$-ball and $c\in \cl{\cF(W)}(\bd X)$,
+let $\cF(W)(X; c) \deq A_\cF(W\times X; c)$.
+\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}[Traditional $n$-categories]
\rm
\label{ex:traditional-n-categories}
@@ -730,7 +746,7 @@
to be the set of all $C$-labeled embedded cell complexes of $X\times F$.
Define $\cC(X; c)$, for $X$ an $n$-ball,
to be the dual Hilbert space $A(X\times F; c)$.
-(See Subsection \ref{sec:constructing-a-tqft}.)
+(See \S\ref{sec:constructing-a-tqft}.)
\end{example}
\noop{
@@ -844,8 +860,8 @@
Note in particular that the space for $k=1$ contains a copy of $\Diff(B^n)$, namely
the embeddings of a ``little" ball with image all of the big ball $B^n$.
\nn{should we warn that the inclusion of this copy of $\Diff(B^n)$ is not a homotopy equivalence?})
-The operad $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad.
-By shrinking the little balls (precomposing them with dilations),
+The operad $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad:
+by shrinking the little balls (precomposing them with dilations),
we see that both operads are homotopic to the space of $k$ framed points
in $B^n$.
It is easy to see that $n$-fold loop spaces $\Omega^n(T)$ have
@@ -876,9 +892,9 @@
also comes from the $\cE\cB_n$ action on $A$.
\nn{should we spell this out?}
-\nn{Should remark that this is just Lurie's topological chiral homology construction
-applied to $n$-balls (check this).
-Hmmm... Does Lurie do both framed and unframed cases?}
+\nn{Should remark that the associated hocolim for manifolds
+is agrees with Lurie's topological chiral homology construction; maybe wait
+until next subsection to say that?}
Conversely, one can show that a topological $A_\infty$ $n$-category $\cC$, where the $k$-morphisms
$\cC(X)$ are trivial (single point) for $k<n$, gives rise to
@@ -887,11 +903,6 @@
\end{example}
-
-
-
-
-%\subsection{From $n$-categories to systems of fields}
\subsection{From balls to manifolds}
\label{ss:ncat_fields} \label{ss:ncat-coend}
In this section we describe how to extend an $n$-category $\cC$ as described above
@@ -913,22 +924,23 @@
We will first define the ``cell-decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$.
An $n$-category $\cC$ provides a functor from this poset to the category of sets,
-and we will define $\cC(W)$ as a suitable colimit
+and we will define $\cl{\cC}(W)$ as a suitable colimit
(or homotopy colimit in the $A_\infty$ case) of this functor.
We'll later give a more explicit description of this colimit.
-In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-manifolds with boundary data),
+In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-balls with boundary data),
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).
-\begin{defn}
-Say that a ``permissible decomposition" of $W$ is a cell decomposition
+Define a {\it permissible decomposition} of $W$ to be a cell decomposition
\[
W = \bigcup_a X_a ,
\]
where each closed top-dimensional cell $X_a$ is an embedded $k$-ball.
+\nn{need to define this more carefully}
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement
of $y$, or write $x \le y$, if each $k$-ball of $y$ is a union of $k$-balls of $x$.
-The category $\cell(W)$ has objects the permissible decompositions of $W$,
+\begin{defn}
+The category (poset) $\cell(W)$ has objects the permissible decompositions of $W$,
and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.
See Figure \ref{partofJfig} for an example.
\end{defn}
@@ -941,15 +953,12 @@
\label{partofJfig}
\end{figure}
-
-
An $n$-category $\cC$ determines
a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets
(possibly with additional structure if $k=n$).
Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls,
and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries
are splittable along this decomposition.
-%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.
\begin{defn}
Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows.
@@ -963,15 +972,20 @@
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 $\cC$.
\end{defn}
-When the $n$-category $\cC$ is enriched in some symmetric monoidal category $(A,\boxtimes)$, and $W$ is a
-closed $n$-manifold, the functor $\psi_{\cC;W}$ has target $A$ and
-we replace the cartesian product of sets appearing in Equation \eqref{eq:psi-C} with the monoidal product $\boxtimes$.
-(Moreover, $\psi_{\cC;W}(x)$ might be a subobject, rather than a subset, of the product.)
-Similar things are true if $W$ is an $n$-manifold with non-empty boundary and we
-fix a field on $\bd W$
-(i.e. fix an element of the colimit associated to $\bd W$).
+If $k=n$ in the above definition and we are enriching in some auxiliary category,
+we need to say a bit more.
+We can rewrite Equation \ref{eq:psi-C} as
+\begin{equation} \label{eq:psi-CC}
+ \psi_{\cC;W}(x) \deq \coprod_\beta \prod_a \cC(X_a; \beta) ,
+\end{equation}
+where $\beta$ runs through labelings of the $k{-}1$-skeleton of the decomposition
+(which are compatible when restricted to the $k{-}2$-skeleton), and $\cC(X_a; \beta)$
+means the subset of $\cC(X_a)$ whose restriction to $\bd X_a$ agress with $\beta$.
+If we are enriching over $\cS$ and $k=n$, then $\cC(X_a; \beta)$ is an object in
+$\cS$ and the coproduct and product in Equation \ref{eq:psi-CC} should be replaced by the approriate
+operations in $\cS$ (e.g. direct sum and tensor product if $\cS$ is Vect).
-Finally, we construct $\cC(W)$ as the appropriate colimit of $\psi_{\cC;W}$.
+Finally, we construct $\cl{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$.
\begin{defn}[System of fields functor]
\label{def:colim-fields}
@@ -990,11 +1004,19 @@
We can specify boundary data $c \in \cl{\cC}(\bdy W)$, and define functors $\psi_{\cC;W,c}$
with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$.
-We now give a more concrete description of the colimit in each case.
-If $\cC$ is enriched over vector spaces, and $W$ is an $n$-manifold,
-we can take the vector space $\cl{\cC}(W,c)$ to be the direct sum over all permissible decompositions of $W$
+We now give more concrete descriptions of the above colimits.
+
+In the non-enriched case (e.g.\ $k<n$), where each $\cC(X_a; \beta)$ is just a set,
+the colimit is
+\[
+ \cl{\cC}(W,c) = \left( \coprod_x \coprod_\beta \prod_a \cC(X_a; \beta) \right) / \sim ,
+\]
+where $x$ runs through decomposition of $W$, and $\sim$ is the obvious equivalence relation
+induced by refinement and gluing.
+If $\cC$ is enriched over vector spaces and $W$ is an $n$-manifold,
+we can take
\begin{equation*}
- \cl{\cC}(W,c) = \left( \bigoplus_x \psi_{\cC;W,c}(x)\right) \big/ K
+ \cl{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) / K
\end{equation*}
where $K$ is the vector space spanned by elements $a - g(a)$, with
$a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x)
@@ -1012,6 +1034,7 @@
where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$.
(Our homological conventions are non-standard: if a complex $U$ is concentrated in degree $0$,
the complex $U[m]$ is concentrated in degree $m$.)
+\nn{if there is a std convention, should we use it? or are we deliberately bucking tradition?}
We endow $\cl{\cC}(W)$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$
summands plus another term using the differential of the simplicial set of $m$-sequences.
More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$
@@ -1021,12 +1044,13 @@
\]
where $d_j(\bar{x}) = (x_0,\dots,x_{j-1},x_{j+1},\dots,x_k)$ and $g: \psi_\cC(x_0)\to \psi_\cC(x_1)$
is the usual gluing map coming from the antirefinement $x_0 \le x_1$.
-\nn{need to say this better}
-\nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which
-combine only two balls at a time; for $n=1$ this version will lead to usual definition
-of $A_\infty$ category}
+%\nn{need to say this better}
+%\nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which
+%combine only two balls at a time; for $n=1$ this version will lead to usual definition
+%of $A_\infty$ category}
We will call $m$ the filtration degree of the complex.
+\nn{is there a more standard term for this?}
We can think of this construction as starting with a disjoint copy of a complex for each
permissible decomposition (filtration degree 0).
Then we glue these together with mapping cylinders coming from gluing maps
@@ -1034,10 +1058,10 @@
Then we kill the extra homology we just introduced with mapping
cylinders between the mapping cylinders (filtration degree 2), and so on.
-$\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}.
+$\cl{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}.
-\todo{This next sentence is circular: these maps are an axiom, not a consequence of anything. -S} It is easy to see that
-there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps
+It is easy to see that
+there are well-defined maps $\cl{\cC}(W)\to\cl{\cC}(\bd W)$, and that these maps
comprise a natural transformation of functors.
\begin{lem}
@@ -1050,8 +1074,6 @@
\end{proof}
\nn{need to finish explaining why we have a system of fields;
-need to say more about ``homological" fields?
-(actions of homeomorphisms);
define $k$-cat $\cC(\cdot\times W)$}
\subsection{Modules}
@@ -1059,17 +1081,12 @@
Next we define plain and $A_\infty$ $n$-category modules.
The definition will be very similar to that of $n$-categories,
but with $k$-balls replaced by {\it marked $k$-balls,} defined below.
-\nn{** need to make sure all revisions of $n$-cat def are also made to module def.}
-\nn{in particular, need to to get rid of the ``hemisphere axiom"}
-%\nn{should they be called $n$-modules instead of just modules? probably not, but worth considering.}
Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary
in the context of an $m{+}1$-dimensional TQFT.
Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$.
This will be explained in more detail as we present the axioms.
-\nn{should also develop $\pi_{\le n}(T, S)$ as a module for $\pi_{\le n}(T)$, where $S\sub T$.}
-
Throughout, we fix an $n$-category $\cC$.
For all but one axiom, it doesn't matter whether $\cC$ is a topological $n$-category or an $A_\infty$ $n$-category.
We state the final axiom, on actions of homeomorphisms, differently in the two cases.
@@ -1088,14 +1105,15 @@
(As with $n$-categories, we will usually omit the subscript $k$.)
-For example, let $\cD$ be the $m{+}1$-dimensional TQFT which assigns to a $k$-manifold $N$ the set
-of maps from $N$ to $T$, modulo homotopy (and possibly linearized) if $k=m$.
+For example, let $\cD$ be the TQFT which assigns to a $k$-manifold $N$ the set
+of maps from $N$ to $T$ (for $k\le m$), modulo homotopy (and possibly linearized) if $k=m$.
Let $W$ be an $(m{-}n{+}1)$-dimensional manifold with boundary.
Let $\cC$ be the $n$-category with $\cC(X) \deq \cD(X\times \bd W)$.
-Let $\cM(B, N) \deq \cD((B\times \bd W)\cup (N\times W))$.
+Let $\cM(B, N) \deq \cD((B\times \bd W)\cup (N\times W))$
+(see Example \ref{ex:maps-with-fiber}).
(The union is along $N\times \bd W$.)
-(If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be
-the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.)
+%(If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be
+%the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.)
\begin{figure}[!ht]
$$\mathfig{.8}{ncat/boundary-collar}$$
@@ -1106,49 +1124,48 @@
\begin{lem}
\label{lem:hemispheres}
-{For each $0 \le k \le n-1$, we have a functor $\cM_k$ from
+{For each $0 \le k \le n-1$, we have a functor $\cl\cM_k$ from
the category of marked $k$-hemispheres and
homeomorphisms to the category of sets and bijections.}
\end{lem}
The proof is exactly analogous to that of Lemma \ref{lem:spheres}, and we omit the details.
We use the same type of colimit construction.
-In our example, let $\cM(H) \deq \cD(H\times\bd W \cup \bd H\times W)$.
+In our example, $\cl\cM(H) = \cD(H\times\bd W \cup \bd H\times W)$.
\begin{module-axiom}[Module boundaries (maps)]
-{For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cM(\bd M)$.
+{For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cl\cM(\bd M)$.
These maps, for various $M$, comprise a natural transformation of functors.}
\end{module-axiom}
-Given $c\in\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$.
+Given $c\in\cl\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$.
If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces),
then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$
and $c\in \cC(\bd M)$.
\begin{lem}[Boundary from domain and range]
-{Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k$-hemisphere ($0\le k\le n-1$),
-$M_i$ is a marked $k$-ball, and $E = M_1\cap M_2$ is a marked $k{-}1$-hemisphere.
+{Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k{-}1$-hemisphere ($1\le k\le n$),
+$M_i$ is a marked $k{-}1$-ball, and $E = M_1\cap M_2$ is a marked $k{-}2$-hemisphere.
Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the
-two maps $\bd: \cM(M_i)\to \cM(E)$.
-Then (axiom) we have an injective map
+two maps $\bd: \cM(M_i)\to \cl\cM(E)$.
+Then we have an injective map
\[
- \gl_E : \cM(M_1) \times_{\cM(E)} \cM(M_2) \hookrightarrow \cM(H)
+ \gl_E : \cM(M_1) \times_{\cl\cM(E)} \cM(M_2) \hookrightarrow \cl\cM(H)
\]
which is natural with respect to the actions of homeomorphisms.}
\end{lem}
Again, this is in exact analogy with Lemma \ref{lem:domain-and-range}.
-Let $\cM(H)_E$ denote the image of $\gl_E$.
-We will refer to elements of $\cM(H)_E$ as ``splittable along $E$" or ``transverse to $E$".
-
+Let $\cl\cM(H)_E$ denote the image of $\gl_E$.
+We will refer to elements of $\cl\cM(H)_E$ as ``splittable along $E$" or ``transverse to $E$".
-\begin{module-axiom}[Module to category restrictions]
+\begin{lem}[Module to category restrictions]
{For each marked $k$-hemisphere $H$ there is a restriction map
-$\cM(H)\to \cC(H)$.
+$\cl\cM(H)\to \cC(H)$.
($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.)
These maps comprise a natural transformation of functors.}
-\end{module-axiom}
+\end{lem}
Note that combining the various boundary and restriction maps above
(for both modules and $n$-categories)
@@ -1215,9 +1232,11 @@
\end{module-axiom}
\begin{module-axiom}[Strict associativity]
-{The composition and action maps above are strictly associative.}
+The composition and action maps above are strictly associative.
\end{module-axiom}
+\nn{should say that this is multifold, not just 3-fold}
+
Note that the above associativity axiom applies to mixtures of module composition,
action maps and $n$-category composition.
See Figure \ref{zzz1b}.
@@ -1251,37 +1270,93 @@
and these various multifold composition maps satisfy an
operad-type strict associativity condition.}
-(The above operad-like structure is analogous to the swiss cheese operad
-\cite{MR1718089}.)
-%\nn{need to double-check that this is true.}
+The above operad-like structure is analogous to the swiss cheese operad
+\cite{MR1718089}.
+
+\medskip
+
+We can define marked pinched products $\pi:E\to M$ of marked balls analogously to the
+plain ball case.
+Note that a marked pinched product can be decomposed into either
+two marked pinched products or a plain pinched product and a marked pinched product.
+\nn{should give figure}
-\begin{module-axiom}[Product/identity morphisms]
-{Let $M$ be a marked $k$-ball and $D$ be a plain $m$-ball, with $k+m \le n$.
-Then we have a map $\cM(M)\to \cM(M\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cM(M)$.
-If $f:M\to M'$ and $\tilde{f}:M\times D \to M'\times D'$ are maps such that the diagram
+\begin{module-axiom}[Product (identity) morphisms]
+For each pinched product $\pi:E\to M$, with $M$ a marked $k$-ball and $E$ a marked
+$k{+}m$-ball ($m\ge 1$),
+there is a map $\pi^*:\cM(M)\to \cM(E)$.
+These maps must satisfy the following conditions.
+\begin{enumerate}
+\item
+If $\pi:E\to M$ and $\pi':E'\to M'$ are marked pinched products, and
+if $f:M\to M'$ and $\tilde{f}:E \to E'$ are maps such that the diagram
\[ \xymatrix{
- M\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & M'\times D' \ar[d]^{\pi} \\
+ E \ar[r]^{\tilde{f}} \ar[d]_{\pi} & E' \ar[d]^{\pi'} \\
M \ar[r]^{f} & M'
} \]
-commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.}
+commutes, then we have
+\[
+ \pi'^*\circ f = \tilde{f}\circ \pi^*.
+\]
+\item
+Product morphisms are compatible with module composition and module action.
+Let $\pi:E\to M$, $\pi_1:E_1\to M_1$, and $\pi_2:E_2\to M_2$
+be pinched products with $E = E_1\cup E_2$.
+Let $a\in \cM(M)$, and let $a_i$ denote the restriction of $a$ to $M_i\sub M$.
+Then
+\[
+ \pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) .
+\]
+Similarly, if $\rho:D\to X$ is a pinched product of plain balls and
+$E = D\cup E_1$, then
+\[
+ \pi^*(a) = \rho^*(a')\bullet \pi_1^*(a_1),
+\]
+where $a'$ is the restriction of $a$ to $D$.
+\item
+Product morphisms are associative.
+If $\pi:E\to M$ and $\rho:D\to E$ are marked pinched products then
+\[
+ \rho^*\circ\pi^* = (\pi\circ\rho)^* .
+\]
+\item
+Product morphisms are compatible with restriction.
+If we have a commutative diagram
+\[ \xymatrix{
+ D \ar@{^(->}[r] \ar[d]_{\rho} & E \ar[d]^{\pi} \\
+ Y \ar@{^(->}[r] & M
+} \]
+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 $k<j$ let
+$\cF(Y)(M)\deq \cF((B\times W) \cup (N\times Y))$.
+If $M = (B, N)$ is a marked $j$-ball and $c\in \cl{\cF(Y)}(\bd M)$ let
+$\cF(Y)(M)\deq A_\cF((B\times W) \cup (N\times Y); c)$.
\end{example}
\begin{example}
+\rm
Suppose $S$ is a topological space, with a subspace $T$.
We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$
for $k<n$ the set $\pi_{\leq n}(S,T)(B,N)$ consists of all continuous maps of pairs
$(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all
such maps modulo homotopies fixed on $\bdy B \setminus N$.
This is a module over the fundamental $n$-category $\pi_{\leq n}(S)$ of $S$, from Example \ref{ex:maps-to-a-space}.
+\end{example}
Modifications corresponding to Examples \ref{ex:maps-to-a-space-with-a-fiber} and
\ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to
Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains.
-\end{example}
\subsection{Modules as boundary labels (colimits for decorated manifolds)}
\label{moddecss}
@@ -1425,7 +1508,7 @@
\label{ss:module-morphisms}
In order to state and prove our version of the higher dimensional Deligne conjecture
-(Section \ref{sec:deligne}),
+(\S\ref{sec:deligne}),
we need to define morphisms of $A_\infty$ $1$-category modules and establish
some of their elementary properties.
@@ -1794,7 +1877,7 @@
of these points labeled by $n$-categories $\cA_i$ ($0\le i\le l$) and the marked points labeled
by $\cA_i$-$\cA_{i+1}$ bimodules $\cM_i$.
(See Figure \ref{feb21c}.)
-To this data we can apply the coend construction as in Subsection \ref{moddecss} above
+To this data we can apply the coend construction as in \S\ref{moddecss} above
to obtain an $\cA_0$-$\cA_l$ $0$-sphere module and, forgetfully, an $n{-}1$-category.
This amounts to a definition of taking tensor products of $0$-sphere module over $n$-categories.
--- a/text/tqftreview.tex Mon Jul 05 10:27:51 2010 -0700
+++ b/text/tqftreview.tex Mon Jul 12 21:46:32 2010 -0600
@@ -16,17 +16,17 @@
A system of fields is very closely related to an $n$-category.
In one direction, Example \ref{ex:traditional-n-categories(fields)}
shows how to construct a system of fields from a (traditional) $n$-category.
-We do this in detail for $n=1,2$ (Subsection \ref{sec:example:traditional-n-categories(fields)})
+We do this in detail for $n=1,2$ (\S\ref{sec:example:traditional-n-categories(fields)})
and more informally for general $n$.
In the other direction,
-our preferred definition of an $n$-category in Section \ref{sec:ncats} is essentially
+our preferred definition of an $n$-category in \S\ref{sec:ncats} is essentially
just a system of fields restricted to balls of dimensions 0 through $n$;
one could call this the ``local" part of a system of fields.
Since this section is intended primarily to motivate
-the blob complex construction of Section \ref{sec:blob-definition},
+the blob complex construction of \S\ref{sec:blob-definition},
we suppress some technical details.
-In Section \ref{sec:ncats} the analogous details are treated more carefully.
+In \S\ref{sec:ncats} the analogous details are treated more carefully.
\medskip
@@ -71,7 +71,7 @@
\end{example}
Now for the rest of the definition of system of fields.
-(Readers desiring a more precise definition should refer to Subsection \ref{ss:n-cat-def}
+(Readers desiring a more precise definition should refer to \S\ref{ss:n-cat-def}
and replace $k$-balls with $k$-manifolds.)
\begin{enumerate}
\item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$,