--- a/text/ncat.tex Mon Jul 12 10:17:52 2010 -0600
+++ b/text/ncat.tex Mon Jul 12 10:17:57 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}
@@ -253,6 +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}
@@ -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
\[
\rho^*\circ\pi^* = (\pi\circ\rho)^* .
\]
@@ -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.
@@ -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{
@@ -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
@@ -1063,8 +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}
@@ -1072,17 +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.
@@ -1101,14 +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]
$$\mathfig{.8}{ncat/boundary-collar}$$
@@ -1119,49 +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)
@@ -1228,9 +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}.
@@ -1264,37 +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 $\bd B \setmin N$.
@@ -1303,19 +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
@@ -1325,7 +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
@@ -1341,10 +1402,19 @@
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
@@ -1438,7 +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.
@@ -1807,7 +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.