--- a/text/ncat.tex Fri Jun 04 17:15:53 2010 -0700
+++ b/text/ncat.tex Fri Jun 04 18:26:04 2010 -0700
@@ -292,6 +292,84 @@
The next axiom is related to identity morphisms, though that might not be immediately obvious.
+\begin{axiom}[Product (identity) morphisms, preliminary version]
+For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$,
+usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$.
+These maps must satisfy the following conditions.
+\begin{enumerate}
+\item
+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
+\[
+ \tilde{f}(a\times D) = f(a)\times D' .
+\]
+\item
+Product morphisms are compatible with gluing (composition) in both factors:
+\[
+ (a'\times D)\bullet(a''\times D) = (a'\bullet a'')\times D
+\]
+and
+\[
+ (a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') .
+\]
+\item
+Product morphisms are associative:
+\[
+ (a\times D)\times D' = a\times (D\times D') .
+\]
+(Here we are implicitly using functoriality and the obvious homeomorphism
+$(X\times D)\times D' \to X\times(D\times D')$.)
+\item
+Product morphisms are compatible with restriction:
+\[
+ \res_{X\times E}(a\times D) = a\times E
+\]
+for $E\sub \bd D$ and $a\in \cC(X)$.
+\end{enumerate}
+\end{axiom}
+
+We will need to strengthen the above preliminary version of the axiom to allow
+for products which are ``pinched" in various ways along their boundary.
+(See Figure xxxx.)
+(The need for a strengthened version will become apparent in appendix \ref{sec:comparing-defs}.)
+Define a {\it pinched product} to be a map
+\[
+ \pi: E\to X
+\]
+such that $E$ is an $m$-ball, $X$ is a $k$-ball ($k<m$), and $\pi$ is locally modeled
+on a standard iterated degeneracy map
+\[
+ d: \Delta^m\to\Delta^k .
+\]
+In other words, \nn{each point has a neighborhood blah blah...}
+(We thank Kevin Costello for suggesting this approach.)
+
+Note that for each interior point $x\in X$, $\pi\inv(x)$ is an $m{-}k$-ball,
+and for for each boundary point $x\in\bd X$, $\pi\inv(x)$ is a ball of dimension
+$l \le m-k$.
+
+It is easy to see that a composition of pinched products is again a pinched product.
+
+A {\it sub pinched product} is a sub-$m$-ball $E'\sub E$ such that the restriction
+$\pi:E'\to \pi(E')$ is again a pinched product.
+A {union} of pinched products is a decomposition $E = \cup_i E_i$
+such that each $E_i\sub E$ is a sub pinched product.
+(See Figure xxxx.)
+
+The product axiom will give a map $\pi^*:\cC(X)\to \cC(E)$ for each pinched product
+$\pi:E\to X$.
+Before stating the axiom, we illustrate it in our two motivating examples of $n$-categories.
+In the case where $\cC(X) = \{f: X\to T\}$, we define $\pi^*(f) = f\circ\pi$.
+In the case where $\cC(X)$ is the set of all labeled embedded cell complexes in $X$,
+taking inverse images under $\pi$ gives a cell decomposition of $E$ in which corresponding cells have the same codimension, and therefore accept the same labels.
+
+\nn{resume revising here; need to rewrite the following pasted axiom}
+
+\addtocounter{axiom}{-1}
\begin{axiom}[Product (identity) morphisms]
For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$,
usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$.
@@ -332,24 +410,16 @@
\end{enumerate}
\end{axiom}
-\nn{need even more subaxioms for product morphisms?}
+
+
+
+
-\nn{Almost certainly we need a little more than the above axiom.
-More specifically, in order to bootstrap our way from the top dimension
-properties of identity morphisms to low dimensions, we need regular products,
-pinched products and even half-pinched products.
-I'm not sure what the best way to cleanly axiomatize the properties of these various
-products is.
-For the moment, I'll assume that all flavors of the product are at
-our disposal, and I'll plan on revising the axioms later.}
+
+
-\nn{current idea for fixing this: make the above axiom a ``preliminary version"
-(as we have already done with some of the other axioms), then state the official
-axiom for maps $\pi: E \to X$ which are almost fiber bundles.
-one option is to restrict E to be a (full/half/not)-pinched product (up to homeo).
-the alternative is to give some sort of local criterion for what's allowed.
-state a gluing axiom for decomps $E = E'\cup E''$ where all three are of the correct type.
-}
+
+\medskip
All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories.
The last axiom (below), concerning actions of