--- a/text/ncat.tex Fri Aug 12 10:00:59 2011 -0600
+++ b/text/ncat.tex Sun Sep 25 14:44:38 2011 -0600
@@ -3,10 +3,10 @@
\def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip}
\def\mmpar#1#2#3{\smallskip\noindent{\bf #1} (#2). {\it #3} \smallskip}
-\section{Disk-like \texorpdfstring{$n$}{n}-categories and their modules}
+\section{\texorpdfstring{$n$}{n}-categories and their modules}
\label{sec:ncats}
-\subsection{Definition of disk-like \texorpdfstring{$n$}{n}-categories}
+\subsection{Definition of \texorpdfstring{$n$}{n}-categories}
\label{ss:n-cat-def}
Before proceeding, we need more appropriate definitions of $n$-categories,
@@ -32,11 +32,11 @@
\medskip
-The axioms for a disk-like $n$-category are spread throughout this section.
-Collecting these together, a disk-like $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms},
+The axioms for an $n$-category are spread throughout this section.
+Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms},
\ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product}, \ref{axiom:extended-isotopies} and \ref{axiom:vcones}.
-For an enriched disk-like $n$-category we add Axiom \ref{axiom:enriched}.
-For an $A_\infty$ disk-like $n$-category, we replace
+For an enriched $n$-category we add Axiom \ref{axiom:enriched}.
+For an $A_\infty$ $n$-category, we replace
Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}.
Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms
@@ -88,7 +88,7 @@
%\nn{need to check whether this makes much difference}
(If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need
to be fussier about corners and boundaries.)
-For each flavor of manifold there is a corresponding flavor of disk-like $n$-category.
+For each flavor of manifold there is a corresponding flavor of $n$-category.
For simplicity, we will concentrate on the case of PL unoriented manifolds.
An ambitious reader may want to keep in mind two other classes of balls.
@@ -814,8 +814,8 @@
\medskip
-This completes the definition of a disk-like $n$-category.
-Next we define enriched disk-like $n$-categories.
+This completes the definition of an $n$-category.
+Next we define enriched $n$-categories.
\medskip
@@ -844,7 +844,7 @@
For convenience, we will also assume that the objects of our auxiliary category are sets with extra structure.
(Otherwise, stating the axioms for identity morphisms becomes more cumbersome.)
-Before stating the revised axioms for a disk-like $n$-category enriched in a distributive monoidal category,
+Before stating the revised axioms for an $n$-category enriched in a distributive monoidal category,
we need a preliminary definition.
Once we have the above $n$-category axioms for $n{-}1$-morphisms, we can define the
category $\bbc$ of {\it $n$-balls with boundary conditions}.
@@ -853,10 +853,10 @@
homeomorphisms $f:X\to X'$ such that $f|_{\bd X}(c) = c'$.
%Let $\pi_0(\bbc)$ denote
-\begin{axiom}[Enriched disk-like $n$-categories]
+\begin{axiom}[Enriched $n$-categories]
\label{axiom:enriched}
Let $\cS$ be a distributive symmetric monoidal category.
-A disk-like $n$-category enriched in $\cS$ satisfies the above $n$-category axioms for $k=0,\ldots,n-1$,
+An $n$-category enriched in $\cS$ satisfies the above $n$-category axioms for $k=0,\ldots,n-1$,
and modifies the axioms for $k=n$ as follows:
\begin{itemize}
\item Morphisms. We have a functor $\cC_n$ from $\bbc$ ($n$-balls with boundary conditions) to $\cS$.
@@ -882,7 +882,7 @@
or more generally an appropriate sort of $\infty$-category,
we can modify the extended isotopy axiom \ref{axiom:extended-isotopies}
to require that families of homeomorphisms act
-and obtain what we shall call an $A_\infty$ disk-like $n$-category.
+and obtain what we shall call an $A_\infty$ $n$-category.
\noop{
We believe that abstract definitions should be guided by diverse collections
@@ -935,7 +935,7 @@
(This is the example most relevant to this paper.)
Then compatibility implies that the action of $C_*(\Homeo(X,c; X', c'))$ agrees with the action
of $C_0(\Homeo(X,c; X', c'))$ coming from Axiom \ref{axiom:morphisms}, so we only require associativity in degree zero.
-And indeed, this is true for our main example of an $A_\infty$ disk-like $n$-category based on the blob construction.
+And indeed, this is true for our main example of an $A_\infty$ $n$-category based on the blob construction.
Stating this sort of compatibility for general $\cS$ and $\cJ$ requires further assumptions,
such as the forgetful functor from $\cS$ to sets having a left adjoint, and $\cS$ having an internal Hom.
@@ -957,7 +957,7 @@
For future reference we make the following definition.
\begin{defn}
-A {\em strict $A_\infty$ disk-like $n$-category} is one in which the actions of Axiom \ref{axiom:families} are strictly associative.
+A {\em strict $A_\infty$ $n$-category} is one in which the actions of Axiom \ref{axiom:families} are strictly associative.
\end{defn}
\noop{
@@ -973,13 +973,13 @@
\medskip
-We define a $j$ times monoidal disk-like $n$-category to be a disk-like $(n{+}j)$-category $\cC$ where
+We define a $j$ times monoidal $n$-category to be an $(n{+}j)$-category $\cC$ where
$\cC(X)$ is a trivial 1-element set if $X$ is a $k$-ball with $k<j$.
See Example \ref{ex:bord-cat}.
\medskip
-The alert reader will have already noticed that our definition of an (ordinary) disk-like $n$-category
+The alert reader will have already noticed that our definition of an (ordinary) $n$-category
is extremely similar to our definition of a system of fields.
There are two differences.
First, for the $n$-category definition we restrict our attention to balls
@@ -988,7 +988,7 @@
invariance in dimension $n$, while in the fields definition we
instead remember a subspace of local relations which contain differences of isotopic fields.
(Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.)
-Thus a system of fields and local relations $(\cF,U)$ determines a disk-like $n$-category $\cC_ {\cF,U}$ simply by restricting our attention to
+Thus a system of fields and local relations $(\cF,U)$ determines an $n$-category $\cC_ {\cF,U}$ simply by restricting our attention to
balls and, at level $n$, quotienting out by the local relations:
\begin{align*}
\cC_{\cF,U}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / U(B) & \text{when $k=n$.}\end{cases}
@@ -1004,7 +1004,7 @@
We also remind the reader of the inductive nature of the definition: All the data for $k{-}1$-morphisms must be in place
before we can describe the data for $k$-morphisms.
-A disk-like $n$-category consists of the following data:
+An $n$-category consists of the following data:
\begin{itemize}
\item functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms});
\item boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary});
@@ -1030,7 +1030,7 @@
\end{itemize}
-\subsection{Examples of disk-like \texorpdfstring{$n$}{n}-categories}
+\subsection{Examples of \texorpdfstring{$n$}{n}-categories}
\label{ss:ncat-examples}
@@ -1162,7 +1162,7 @@
where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary,
and $C_*$ denotes singular chains.
Alternatively, if we take the $n$-morphisms to be simply $\Maps_c(X \to T)$,
-we get an $A_\infty$ disk-like $n$-category enriched over spaces.
+we get an $A_\infty$ $n$-category enriched over spaces.
\end{example}
See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to
@@ -1172,7 +1172,7 @@
\rm
\label{ex:blob-complexes-of-balls}
Fix an $n{-}k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$.
-We will define an $A_\infty$ disk-like $k$-category $\cC$.
+We will define an $A_\infty$ $k$-category $\cC$.
When $X$ is an $m$-ball, with $m<k$, define $\cC(X) = \cE(X\times F)$.
When $X$ is a $k$-ball,
define $\cC(X; c) = \bc^\cE_*(X\times F; c)$
@@ -1180,17 +1180,17 @@
\end{example}
This example will be used in Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product.
-Notice that with $F$ a point, the above example is a construction turning an ordinary disk-like
-$n$-category $\cC$ into an $A_\infty$ disk-like $n$-category.
+Notice that with $F$ a point, the above example is a construction turning an ordinary
+$n$-category $\cC$ into an $A_\infty$ $n$-category.
We think of this as providing a ``free resolution"
-of the ordinary disk-like $n$-category.
+of the ordinary $n$-category.
%\nn{say something about cofibrant replacements?}
In fact, there is also a trivial, but mostly uninteresting, way to do this:
we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$,
and take $\CD{B}$ to act trivially.
-Beware that the ``free resolution" of the ordinary disk-like $n$-category $\pi_{\leq n}(T)$
-is not the $A_\infty$ disk-like $n$-category $\pi^\infty_{\leq n}(T)$.
+Beware that the ``free resolution" of the ordinary $n$-category $\pi_{\leq n}(T)$
+is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$.
It's easy to see that with $n=0$, the corresponding system of fields is just
linear combinations of connected components of $T$, and the local relations are trivial.
There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$.
@@ -1234,7 +1234,7 @@
Let $A$ be an $\cE\cB_n$-algebra.
Note that this implies a $\Diff(B^n)$ action on $A$,
since $\cE\cB_n$ contains a copy of $\Diff(B^n)$.
-We will define a strict $A_\infty$ disk-like $n$-category $\cC^A$.
+We will define a strict $A_\infty$ $n$-category $\cC^A$.
(We enrich in topological spaces, though this could easily be adapted to, say, chain complexes.)
If $X$ is a ball of dimension $k<n$, define $\cC^A(X)$ to be a point.
In other words, the $k$-morphisms are trivial for $k<n$.
@@ -1247,12 +1247,12 @@
to each morphism of $J$ we associate a morphism coming from the $\cE\cB_n$ action on $A$.
Alternatively and more simply, we could define $\cC^A(X)$ to be
$\Diff(B^n\to X)\times A$ modulo the diagonal action of $\Diff(B^n)$.
-The remaining data for the $A_\infty$ disk-like $n$-category
+The remaining data for the $A_\infty$ $n$-category
--- composition and $\Diff(X\to X')$ action ---
also comes from the $\cE\cB_n$ action on $A$.
%\nn{should we spell this out?}
-Conversely, one can show that a strict $A_\infty$ disk-like $n$-category $\cC$, where the $k$-morphisms
+Conversely, one can show that a disk-like strict $A_\infty$ $n$-category $\cC$, where the $k$-morphisms
$\cC(X)$ are trivial (single point) for $k<n$, gives rise to
an $\cE\cB_n$-algebra.
%\nn{The paper is already long; is it worth giving details here?}
@@ -1277,19 +1277,19 @@
\subsection{From balls to manifolds}
\label{ss:ncat_fields} \label{ss:ncat-coend}
-In this section we show how to extend a disk-like $n$-category $\cC$ as described above
+In this section we show how to extend an $n$-category $\cC$ as described above
(of either the ordinary or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$.
This extension is a certain colimit, and the arrow in the notation is intended as a reminder of this.
-In the case of ordinary disk-like $n$-categories, this construction factors into a construction of a
+In the case of ordinary $n$-categories, this construction factors into a construction of a
system of fields and local relations, followed by the usual TQFT definition of a
vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}.
-For an $A_\infty$ disk-like $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead.
-Recall that we can take an ordinary disk-like $n$-category $\cC$ and pass to the ``free resolution",
-an $A_\infty$ disk-like $n$-category $\bc_*(\cC)$, by computing the blob complex of balls
+For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead.
+Recall that we can take an ordinary $n$-category $\cC$ and pass to the ``free resolution",
+an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls
(recall Example \ref{ex:blob-complexes-of-balls} above).
We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant
-for a manifold $M$ associated to this $A_\infty$ disk-like $n$-category is actually the
+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 $\cC$.
Recall that we've already anticipated this construction Subsection \ref{ss:n-cat-def},
@@ -1299,11 +1299,11 @@
\medskip
We will first define the {\it decomposition poset} $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$.
-A disk-like $n$-category $\cC$ provides a functor from this poset to the category of sets,
+An $n$-category $\cC$ provides a functor from this poset to the category of sets,
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 disk-like $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain
+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
@@ -1354,7 +1354,7 @@
\label{partofJfig}
\end{figure}
-A disk-like $n$-category $\cC$ determines
+An $n$-category $\cC$ determines
a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets
(possibly with additional structure if $k=n$).
Let $x = \{X_a\}$ be a permissible decomposition of $W$ (i.e.\ object of $\cD(W)$).
@@ -1422,14 +1422,14 @@
\begin{defn}[System of fields functor]
\label{def:colim-fields}
-If $\cC$ is a disk-like $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$.
+If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$.
That is, for each decomposition $x$ there is a map
$\psi_{\cC;W}(x)\to \cl{\cC}(W)$, these maps are compatible with the refinement maps
above, and $\cl{\cC}(W)$ is universal with respect to these properties.
\end{defn}
\begin{defn}[System of fields functor, $A_\infty$ case]
-When $\cC$ is an $A_\infty$ disk-like $n$-category, $\cl{\cC}(W)$ for $W$ a $k$-manifold with $k < n$
+When $\cC$ is an $A_\infty$ $n$-category, $\cl{\cC}(W)$ for $W$ a $k$-manifold with $k < n$
is defined as above, as the colimit of $\psi_{\cC;W}$.
When $W$ is an $n$-manifold, the chain complex $\cl{\cC}(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$.
\end{defn}
@@ -1605,16 +1605,16 @@
\subsection{Modules}
\label{sec:modules}
-Next we define ordinary and $A_\infty$ disk-like $n$-category modules.
-The definition will be very similar to that of disk-like $n$-categories,
+Next we define ordinary 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.
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 disk-like $n$-category associated to $\bd W$.
+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.
-Throughout, we fix a disk-like $n$-category $\cC$.
+Throughout, we fix an $n$-category $\cC$.
For all but one axiom, it doesn't matter whether $\cC$ is an ordinary $n$-category or an $A_\infty$ $n$-category.
We state the final axiom, regarding actions of homeomorphisms, differently in the two cases.
@@ -1670,7 +1670,7 @@
Given $c\in\cl\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$.
-If the disk-like $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces),
+If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces),
then for each marked $n$-ball $M=(B,N)$ and $c\in \cC(\bd B \setminus N)$, the set $\cM(M; c)$ should be an object in that category.
\begin{lem}[Boundary from domain and range]
@@ -1903,7 +1903,7 @@
\end{enumerate}
\end{module-axiom}
-As in the disk-like $n$-category definition, once we have product morphisms we can define
+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
@@ -1916,7 +1916,7 @@
\medskip
There are two alternatives for the next axiom, according whether we are defining
-modules for ordinary or $A_\infty$ disk-like $n$-categories.
+modules for ordinary $n$-categories or $A_\infty$ $n$-categories.
In the ordinary case we require
\begin{module-axiom}[\textup{\textbf{[ordinary version]}} Extended isotopy invariance in dimension $n$]
@@ -1949,14 +1949,14 @@
\medskip
-Note that the above axioms imply that a disk-like $n$-category module has the structure
-of a disk-like $n{-}1$-category.
+Note that the above axioms imply that an $n$-category module has the structure
+of an $n{-}1$-category.
More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$,
where $X$ is a $k$-ball and in the product $X\times J$ we pinch
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 $J$.)
-Then $\cE$ has the structure of a disk-like $n{-}1$-category.
+Then $\cE$ has the structure of an $n{-}1$-category.
All marked $k$-balls are homeomorphic, unless $k = 1$ and our manifolds
are oriented or Spin (but not unoriented or $\text{Pin}_\pm$).
@@ -1968,12 +1968,12 @@
\medskip
-We now give some examples of modules over ordinary and $A_\infty$ disk-like $n$-categories.
+We now give some examples of modules over ordinary and $A_\infty$ $n$-categories.
\begin{example}[Examples from TQFTs]
\rm
Continuing Example \ref{ex:ncats-from-tqfts}, with $\cF$ a TQFT, $W$ an $n{-}j$-manifold,
-and $\cF(W)$ the disk-like $j$-category associated to $W$.
+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
@@ -1986,7 +1986,7 @@
\rm
In the previous example, we can instead define
$\cF(Y)(M)\deq \bc_*((B\times W) \cup (N\times Y), c; \cF)$ (when $\dim(M) = n$)
-and get a module for the $A_\infty$ disk-like $n$-category associated to $\cF$ as in
+and get a module for the $A_\infty$ $n$-category associated to $\cF$ as in
Example \ref{ex:blob-complexes-of-balls}.
\end{example}
@@ -2011,7 +2011,7 @@
\subsection{Modules as boundary labels (colimits for decorated manifolds)}
\label{moddecss}
-Fix an ordinary or $A_\infty$ disk-like $n$-category $\cC$.
+Fix an ordinary $n$-category or $A_\infty$ $n$-category $\cC$.
Let $W$ be a $k$-manifold ($k\le n$),
let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$,
and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$.
@@ -2067,19 +2067,19 @@
$\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold
$D\times Y_i \sub \bd(D\times W)$.
It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$
-has the structure of a disk-like $n{-}k$-category.
+has the structure of an $n{-}k$-category.
\medskip
We will use a simple special case of the above
construction to define tensor products
of modules.
-Let $\cM_1$ and $\cM_2$ be modules for a disk-like $n$-category $\cC$.
+Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$.
(If $k=1$ and our manifolds are oriented, then one should be
a left module and the other a right module.)
Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$.
Define the tensor product $\cM_1 \tensor \cM_2$ to be the
-disk-like $n{-}1$-category associated as above to $J$ with its boundary labeled by $\cM_1$ and $\cM_2$.
+$n{-}1$-category associated as above to $J$ with its boundary labeled by $\cM_1$ and $\cM_2$.
This of course depends (functorially)
on the choice of 1-ball $J$.