
diff -r 38ceade5cc5d -r b51fcceb1d57 text/ncat.tex
--- a/text/ncat.tex	Mon Jul 20 17:37:50 2009 +0000
+++ b/text/ncat.tex	Mon Jul 20 22:50:59 2009 +0000
@@ -143,5 +143,50 @@
 If $k < n$ we require that $\gl_Y$ is injective.
 (For $k=n$, see below.)}
 
+\xxpar{Strict associativity:}
+{The composition (gluing) maps above are strictly associative.
+It follows that given a decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball
+into small $k$-balls, there is a well-defined
+map from an appropriate subset of $\cC(B_1)\times\cdots\times\cC(B_m)$ to $\cC(B)$,
+and these various $m$-fold composition maps satisfy an
+operad-type associativity condition.}
+
+\nn{above maybe needs some work}
+
+The next axiom is related to identity morphisms, though that might not be immediately obvious.
+
+\xxpar{Product (identity) morphisms:}
+{Let $X$ be homeomorphic to a $k$-ball and $D$ be homeomorphic to an $m$-ball, with $k+m \le n$.
+Then we have a map $\cC(X)\to \cC(X\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$.
+If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram
+\[ \xymatrix{
+	X\times D \ar[r]^{\tilde{f}} \ar[d]^{\pi} & X'\times D' \ar[d]^{\pi} \\
+	X \ar[r]^{f} & X'
+} \]
+commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.}
+
+\nn{Need to say something about compatibility with gluing (of both $X$ and $D$) above.}
+
+All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories.
+The last axiom (below), concerning actions of 
+homeomorphisms in the top dimension $n$, distinguishes the two cases.
+
+We start with the plain $n$-category case.
+
+\xxpar{Isotopy invariance in dimension $n$ (preliminary version):}
+{Let $X$ be homeomorphic to the $n$-ball and $f: X\to X$ be a homeomorphism which restricts
+to the identity on $\bd X$ and is isotopic (rel boundary) to the identity.
+Then $f(a) = a$ for all $a\in \cC(X)$.}
 
 
+
+
+
+\medskip
+
+\hrule
+
+\medskip
+
+\nn{to be continued...}
+




