
diff -r 1eab7b40e897 -r a2ff2d278b97 text/ncat.tex
--- a/text/ncat.tex	Sun Jan 10 20:48:09 2010 +0000
+++ b/text/ncat.tex	Wed Jan 27 18:33:59 2010 +0000
@@ -1,6 +1,7 @@
 %!TEX root = ../blob1.tex
 
 \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{$n$-categories}
 \label{sec:ncats}
@@ -722,12 +723,11 @@
 
 Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations,
 a.k.a.\ actions).
-The definition will be very similar to that of $n$-categories.
+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{should they be called $n$-modules instead of just modules?  probably not, but worth considering.}
 
-\nn{** resume revising here}
-
 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 $\bd W$.
@@ -741,7 +741,7 @@
 A homeomorphism between marked $k$-balls is a homeomorphism of balls which
 restricts to a homeomorphism of markings.
 
-\xxpar{Module morphisms}
+\mmpar{Module axiom 1}{Module morphisms}
 {For each $0 \le k \le n$, we have a functor $\cM_k$ from 
 the category of marked $k$-balls and 
 homeomorphisms to the category of sets and bijections.}
@@ -764,14 +764,14 @@
 Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$.
 Call such a thing a {marked $k{-}1$-hemisphere}.
 
-\xxpar{Module boundaries, part 1:}
+\mmpar{Module axiom 2}{Module boundaries (hemispheres)}
 {For each $0 \le k \le n-1$, we have a functor $\cM_k$ from 
 the category of marked $k$-hemispheres and 
 homeomorphisms to the category of sets and bijections.}
 
 In our example, let $\cM(H) \deq \cD(H\times\bd W \cup \bd H\times W)$.
 
-\xxpar{Module boundaries, part 2:}
+\mmpar{Module axiom 3}{Module boundaries (maps)}
 {For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cM(\bd M)$.
 These maps, for various $M$, comprise a natural transformation of functors.}
 
@@ -781,14 +781,14 @@
 then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$
 and $c\in \cC(\bd M)$.
 
-\xxpar{Module domain $+$ range $\to$ boundary:}
+\mmpar{Module axiom 4}{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 $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the 
 two maps $\bd: \cM(M_i)\to \cM(E)$.
 Then (axiom) we have an injective map
 \[
-	\gl_E : \cM(M_1) \times_{\cM(E)} \cM(M_2) \to \cM(H)
+	\gl_E : \cM(M_1) \times_{\cM(E)} \cM(M_2) \hookrightarrow \cM(H)
 \]
 which is natural with respect to the actions of homeomorphisms.}
 
@@ -796,7 +796,7 @@
 We will refer to elements of $\cM(H)_E$ as ``splittable along $E$" or ``transverse to $E$". 
 
 
-\xxpar{Axiom yet to be named:}
+\mmpar{Module axiom 5}{Module to category restrictions}
 {For each marked $k$-hemisphere $H$ there is a restriction map
 $\cM(H)\to \cC(H)$.  
 ($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.)




