
diff -r 2a8aecc675c7 -r 0e71da01b195 text/ncat.tex
--- a/text/ncat.tex	Sun Feb 21 22:49:18 2010 +0000
+++ b/text/ncat.tex	Sun Feb 21 23:27:38 2010 +0000
@@ -1111,9 +1111,17 @@
 (For $n=1$ they are precisely bimodules in the usual, uncategorified sense.)
 Define a 0-marked $k$-ball $(X, M)$, $1\le k \le n$, to be a pair homeomorphic to the standard
 $(B^k, B^{k-1})$, where $B^{k-1}$ is properly embedded in $B^k$.
-See Figure xxxx.
+See Figure \ref{feb21a}.
 Another way to say this is that $(X, M)$ is homeomorphic to $B^{k-1}\times([-1,1], \{0\})$.
 
+\begin{figure}[!ht]
+\begin{equation*}
+\mathfig{.85}{tempkw/feb21a}
+\end{equation*}
+\caption{0-marked 1-ball and 0-marked 2-ball}
+\label{feb21a}
+\end{figure}
+
 0-marked balls can be cut into smaller balls in various ways.
 These smaller balls could be 0-marked or plain.
 We can also take the boundary of a 0-marked ball, which is 0-marked sphere.
@@ -1146,20 +1154,36 @@
 \]
 The product is pinched over the boundary of $J$.
 $\cD$ breaks into ``blocks" according to the restrictions to the pinched points of $X\times J$
-(see Figure xxxx).
+(see Figure \ref{feb21b}).
 These restrictions are 0-morphisms $(a, b)$ of $\cA$ and $\cB$.
 
+\begin{figure}[!ht]
+\begin{equation*}
+\mathfig{1}{tempkw/feb21b}
+\end{equation*}
+\caption{The pinched product $X\times J$}
+\label{feb21b}
+\end{figure}
+
 More generally, consider an interval with interior marked points, and with the complements
 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 xxxx.)
+(See Figure \ref{feb21c}.)
 To this data we can apply to coend construction as in Subsection \ref{moddecss} above
 to obtain an $\cA_0$-$\cA_l$ bimodule and, forgetfully, an $n{-}1$-category.
 This amounts to a definition of taking tensor products of bimodules over $n$-categories.
 
+\begin{figure}[!ht]
+\begin{equation*}
+\mathfig{1}{tempkw/feb21c}
+\end{equation*}
+\caption{Marked and labeled 1-manifolds}
+\label{feb21c}
+\end{figure}
+
 We could also similarly mark and label a circle, obtaining an $n{-}1$-category
 associated to the marked and labeled circle.
-(See Figure xxxx.)
+(See Figure \ref{feb21c}.)
 If the circle is divided into two intervals, we can think of this $n{-}1$-category
 as the 2-ended tensor product of the two bimodules associated to the two intervals.
 
@@ -1171,7 +1195,7 @@
 
 Equivalently, we can define 1-sphere modules in terms of 1-marked $k$-balls, $2\le k\le n$.
 Fix a marked (and labeled) circle $S$.
-Let $C(S)$ denote the cone of $S$, a marked 2-ball (Figure xxxx).
+Let $C(S)$ denote the cone of $S$, a marked 2-ball (Figure \ref{feb21d}).
 \nn{I need to make up my mind whether marked things are always labeled too.
 For the time being, let's say they are.}
 A 1-marked $k$-ball is anything homeomorphic to $B^j \times C(S)$, $0\le j\le n-2$, 
@@ -1180,6 +1204,14 @@
 smaller 1-marked $k$-balls or the product of an unmarked ball with a marked interval.
 We now proceed as in the above module definitions.
 
+\begin{figure}[!ht]
+\begin{equation*}
+\mathfig{.4}{tempkw/feb21d}
+\end{equation*}
+\caption{Cone on a marked circle}
+\label{feb21d}
+\end{figure}
+
 A $n$-category 1-sphere module is, among other things, an $n{-}2$-category $\cD$ with
 \[
 	\cD(X) \deq \cM(X\times C(S)) .




