
diff -r 9c908b698da5 -r 3b228545d9bb text/ncat.tex
--- a/text/ncat.tex	Fri Nov 06 21:57:13 2009 +0000
+++ b/text/ncat.tex	Sat Nov 07 15:23:53 2009 +0000
@@ -143,7 +143,7 @@
 
 \xxpar{Domain $+$ range $\to$ boundary:}
 {Let $S = B_1 \cup_E B_2$, where $S$ is a $k$-sphere ($0\le k\le n-1$),
-$B_i$ is a $k$-ball, and $E = B_1\cap B_2$ is a $k{-}1$-sphere.
+$B_i$ is a $k$-ball, and $E = B_1\cap B_2$ is a $k{-}1$-sphere (Figure \ref{blah3}).
 Let $\cC(B_1) \times_{\cC(E)} \cC(B_2)$ denote the fibered product of the 
 two maps $\bd: \cC(B_i)\to \cC(E)$.
 Then (axiom) we have an injective map
@@ -152,6 +152,10 @@
 \]
 which is natural with respect to the actions of homeomorphisms.}
 
+\begin{figure}[!ht]
+$$\mathfig{.4}{tempkw/blah3}$$
+\caption{Combining two balls to get a full boundary}\label{blah3}\end{figure}
+
 Note that we insist on injectivity above.
 
 Let $\cC(S)_E$ denote the image of $\gl_E$.
@@ -175,7 +179,7 @@
 
 \xxpar{Composition:}
 {Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$)
-and $Y = B_1\cap B_2$ is a $k{-}1$-ball.
+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 $E$.
 We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$.
@@ -189,9 +193,17 @@
 If $k < n$ we require that $\gl_Y$ is injective.
 (For $k=n$, see below.)}
 
+\begin{figure}[!ht]
+$$\mathfig{.4}{tempkw/blah5}$$
+\caption{From two balls to one ball}\label{blah5}\end{figure}
+
 \xxpar{Strict associativity:}
 {The composition (gluing) maps above are strictly associative.}
 
+\begin{figure}[!ht]
+$$\mathfig{.65}{tempkw/blah6}$$
+\caption{An example of strict associativity}\label{blah6}\end{figure}
+
 Notation: $a\bullet b \deq \gl_Y(a, b)$ and/or $a\cup b \deq \gl_Y(a, b)$.
 In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ 
 a {\it restriction} map and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$.
@@ -212,7 +224,11 @@
 map from an appropriate subset (like a fibered product) 
 of $\cC(B_1)\times\cdots\times\cC(B_m)$ to $\cC(B)$,
 and these various $m$-fold composition maps satisfy an
-operad-type strict associativity condition.}
+operad-type strict associativity condition (Figure \ref{blah7}).}
+
+\begin{figure}[!ht]
+$$\mathfig{.8}{tempkw/blah7}$$
+\caption{Operadish composition and associativity}\label{blah7}\end{figure}
 
 The next axiom is related to identity morphisms, though that might not be immediately obvious.
 




