--- a/text/ncat.tex Sun Dec 11 23:03:27 2011 -0800
+++ b/text/ncat.tex Sun Dec 11 23:03:37 2011 -0800
@@ -256,7 +256,7 @@
\begin{axiom}[Composition]
\label{axiom:composition}
-Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$)
+Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($1\le k\le n$)
and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}).
Let $E = \bd Y$, which is a $k{-}2$-sphere.
Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$.
@@ -1275,9 +1275,9 @@
This last example generalizes Lemma \ref{lem:ncat-from-fields} above which produced an $n$-category from an $n$-dimensional system of fields and local relations. Taking $W$ to be the point recovers that statement.
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.
+which definition of a ``traditional $n$-category with strong duality" 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}
@@ -1368,7 +1368,7 @@
%\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.
+and let $\CH{B}$ act trivially.
Beware that the ``free resolution" of the ordinary $n$-category $\pi_{\leq n}(T)$
is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$.
@@ -1571,7 +1571,7 @@
along $\bd Y_1$ and $\bd Y'_1$, and that the restrictions to $\cl\cC(Y_1)$ and $\cl\cC(Y'_1)$ agree
(with respect to the identification of $Y_1$ and $Y'_1$ provided by the gluing map).
The $i$-th condition is defined similarly.
-Note that these conditions depend on the boundaries of elements of $\prod_a \cC(X_a)$.
+Note that these conditions depend only on the boundaries of elements of $\prod_a \cC(X_a)$.
We define $\psi_{\cC;W}(x)$ to be the subset of $\prod_a \cC(X_a)$ which satisfies the
above conditions for all $i$ and also all
@@ -1729,11 +1729,11 @@
\medskip
$\cl{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds.
-Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}.
+Restricting to $k$-spheres, we have now proved Lemma \ref{lem:spheres}.
\begin{lem}
\label{lem:colim-injective}
-Let $W$ be a manifold of dimension less than $n$. Then for each
+Let $W$ be a manifold of dimension $j<n$. Then for each
decomposition $x$ of $W$ the natural map $\psi_{\cC;W}(x)\to \cl{\cC}(W)$ is injective.
\end{lem}
\begin{proof}
@@ -1796,7 +1796,7 @@
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 $\bd W$.
+Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$ (see Example \ref{ex:ncats-from-tqfts}).
This will be explained in more detail as we present the axioms.
Throughout, we fix an $n$-category $\cC$.
@@ -1818,12 +1818,13 @@
(As with $n$-categories, we will usually omit the subscript $k$.)
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 $k=m$.
+of maps from $N$ to $T$ (for $k\le m$), modulo homotopy (and possibly linearized) if $k=m$
+(see Example \ref{ex:maps-with-fiber}).
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 $\cM(B, N) \deq \cD((B\times \bd W)\cup (N\times W))$
-(see Example \ref{ex:maps-with-fiber}).
+Let $\cM(B, N) \deq \cD((B\times \bd W)\cup (N\times W))$.
(The union is along $N\times \bd W$.)
+See Figure \ref{blah15}.
%(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 $N\times \bd W$.)
@@ -1839,7 +1840,7 @@
\begin{lem}
\label{lem:hemispheres}
-{For each $0 \le k \le n-1$, we have a functor $\cl\cM_k$ from
+{For each $1 \le k \le n$, we have a functor $\cl\cM_{k-1}$ from
the category of marked $k$-hemispheres and
homeomorphisms to the category of sets and bijections.}
\end{lem}
@@ -1870,7 +1871,7 @@
\]
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}, and illustrated in Figure \ref{fig:module-boundary}.
+This is in exact analogy with Lemma \ref{lem:domain-and-range}, and illustrated in Figure \ref{fig:module-boundary}.
\begin{figure}[t]
\begin{equation*}
\begin{tikzpicture}[baseline=0]
@@ -1942,7 +1943,7 @@
First, we can compose two module morphisms to get another module morphism.
\begin{module-axiom}[Module composition]
-{Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls (with $0\le k\le n$)
+{Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls (with $2\le k\le n$)
and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball.
Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere.
Note that each of $M$, $M_1$ and $M_2$ has its boundary split into two marked $k{-}1$-balls by $E$.
@@ -1963,7 +1964,7 @@
We'll call this the action map to distinguish it from the other kind of composition.
\begin{module-axiom}[$n$-category action]
-{Let $M = X \cup_Y M'$, where $M$ and $M'$ are marked $k$-balls ($0\le k\le n$),
+{Let $M = X \cup_Y M'$, where $M$ and $M'$ are marked $k$-balls ($1\le k\le n$),
$X$ is a plain $k$-ball,
and $Y = X\cap M'$ is a $k{-}1$-ball.
Let $E = \bd Y$, which is a $k{-}2$-sphere.
@@ -2153,8 +2154,8 @@
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 $\cC$.
-In our example, elements $a$ of $\cM(M)$ maps to $T$, and $\pi^*(a)$ is the pullback of
-$a$ along a map associated to $\pi$.
+In our example, elements $a$ of $\cM(M)$ are maps to $T$, and $\pi^*(a)$ is the pullback of
+$a$ along the map associated to $\pi$.
\medskip
@@ -2292,7 +2293,7 @@
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 $Y_i$.
+and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each $Y_i$.
We will define a set $\cC(W, \cN)$ using a colimit construction very similar to
the one appearing in \S \ref{ss:ncat_fields} above.
@@ -2330,7 +2331,7 @@
\]
such that the restrictions to the various pieces of shared boundaries amongst the
$X_a$ and $M_{ib}$ all agree.
-(That is, the fibered product over the boundary restriction maps.)
+%(That is, the fibered product over the boundary restriction maps.)
If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$
via the gluing (composition or action) maps from $\cC$ and the $\cN_i$.
@@ -2374,7 +2375,7 @@
additional data.
More specifically, let $\cX$ and $\cY$ be $\cC$ modules, i.e.\ collections of functors
-$\{\cX_k\}$ and $\{\cY_k\}$, for $0\le k\le n$, from marked $k$-balls to sets
+$\{\cX_k\}$ and $\{\cY_k\}$, for $1\le k\le n$, from marked $k$-balls to sets
as in Module Axiom \ref{module-axiom-funct}.
A morphism $g:\cX\to\cY$ is a collection of natural transformations $g_k:\cX_k\to\cY_k$
satisfying:
@@ -2447,10 +2448,12 @@
In this subsection we define $n{+}1$-categories $\cS$ of ``sphere modules".
The objects are $n$-categories, the $k$-morphisms are $k{-}1$-sphere modules for $1\le k \le n$,
and the $n{+}1$-morphisms are intertwiners.
-With future applications in mind, we treat simultaneously the big category
+With future applications in mind, we treat simultaneously the big $n{+}1$-category
of all $n$-categories and all sphere modules and also subcategories thereof.
-When $n=1$ this is closely related to familiar $2$-categories consisting of
-algebras, bimodules and intertwiners (or a subcategory of that).
+When $n=1$ this is closely related to the familiar $2$-category consisting of
+algebras, bimodules and intertwiners, or a subcategory of that.
+(More generally, we can replace algebras with linear 1-categories.)
+The ``bi" in ``bimodule" corresponds to the fact that a 0-sphere consists of two points.
The sphere module $n{+}1$-category is a natural generalization of the
algebra-bimodule-intertwiner 2-category to higher dimensions.
@@ -2462,13 +2465,13 @@
\medskip
-While it is appropriate to call an $S^0$ module a bimodule,
-this is much less true for higher dimensional spheres,
-so we prefer the term ``sphere module" for the general case.
+%While it is appropriate to call an $S^0$ module a bimodule,
+%this is much less true for higher dimensional spheres,
+%so we prefer the term ``sphere module" for the general case.
For simplicity, we will assume that $n$-categories are enriched over $\c$-vector spaces.
-The $0$- through $n$-dimensional parts of $\cS$ are various sorts of modules, and we describe
+The $1$- through $n$-dimensional parts of $\cS$ are various sorts of modules, and we describe
these first.
The $n{+}1$-dimensional part of $\cS$ consists of intertwiners
of $1$-category modules associated to decorated $n$-balls.
@@ -2703,9 +2706,10 @@
The only requirement is that each $k$-sphere module be a module for a $k$-sphere $n{-}k$-category
constructed out of labels taken from $L_j$ for $j<k$.
-We remind the reader again that $\cS = \cS_{\{L_i\}, \{z_Y\}}$ depends on
+%We remind the reader again that $\cS = \cS_{\{L_i\}, \{z_Y\}}$ depends on
+We remind the reader again that $\cS$ depends on
the choice of $L_i$ above as well as the choice of
-families of inner products below.
+families of inner products described below.
We now define $\cS(X)$, for $X$ a ball of dimension at most $n$, to be the set of all
cell-complexes $K$ embedded in $X$, with the codimension-$j$ parts of $(X, K)$ labeled
@@ -2727,7 +2731,7 @@
$n{+}1$-morphisms will require some effort and combinatorial topology, as well as additional
duality assumptions on the lower morphisms.
These are required because we define the spaces of $n{+}1$-morphisms by
-making arbitrary choices of incoming and outgoing boundaries for each $n$-ball.
+making arbitrary choices of incoming and outgoing boundaries for each $n{+}1$-ball.
The additional duality assumptions are needed to prove independence of our definition from these choices.
Let $X$ be an $n{+}1$-ball, and let $c$ be a decoration of its boundary
@@ -3246,7 +3250,7 @@
\end{tikzpicture}
$$
-\caption{intertwiners for a Morita equivalence}\label{morita-fig-2}
+\caption{Intertwiners for a Morita equivalence}\label{morita-fig-2}
\end{figure}
shows the intertwiners we need.
Each decorated 2-ball in that figure determines a representation of the 1-category associated to the decorated circle
@@ -3316,7 +3320,7 @@
\caption{Identities for intertwiners}\label{morita-fig-3}
\end{figure}
Each line shows a composition of two intertwiners which we require to be equal to the identity intertwiner.
-The modules corresponding leftmost and rightmost disks in the figure can be identified via the obvious isotopy.
+The modules corresponding to the leftmost and rightmost disks in the figure can be identified via the obvious isotopy.
For general $n$, we start with an $n$-category 0-sphere module $\cM$ which is the data for the 1-dimensional
part of the Morita equivalence.