--- a/blob to-do Sun Jun 19 17:07:48 2011 -0600
+++ b/blob to-do Sun Jun 19 17:31:34 2011 -0600
@@ -7,8 +7,6 @@
* ** new material in colimit section needs a proof-read
-* should probably allow product things \pi^*(b) to be defined only when b is appropriately splittable
-
* framings and duality -- work out what's going on! (alternatively, vague-ify current statement)
* make sure we are clear that boundary = germ
--- a/blob_changes_v3 Sun Jun 19 17:07:48 2011 -0600
+++ b/blob_changes_v3 Sun Jun 19 17:31:34 2011 -0600
@@ -28,6 +28,7 @@
- extended the lemmas of Appendix B (about adapting families of homeomorphisms to open covers) to the topological category
- modified families-of-homeomorphisms-action axiom for A-infinity n-categories, and added discussion of alternatives
- added n-cat axiom for existence of splittings
+- added transversality requirement to product morphism axiom
--- a/text/ncat.tex Sun Jun 19 17:07:48 2011 -0600
+++ b/text/ncat.tex Sun Jun 19 17:31:34 2011 -0600
@@ -492,7 +492,11 @@
\caption{Five examples of unions of pinched products}\label{pinched_prod_unions}
\end{figure}
-The product axiom will give a map $\pi^*:\cC(X)\to \cC(E)$ for each pinched product
+Note that $\bd X$ has a (possibly trivial) subdivision according to
+the dimension of $\pi\inv(x)$, $x\in \bd X$.
+Let $\cC(X)\trans{}$ denote the morphisms which are splittable along this subdivision.
+
+The product axiom will give a map $\pi^*:\cC(X)\trans{}\to \cC(E)$ for each pinched product
$\pi:E\to X$.
Morphisms in the image of $\pi^*$ will be called product morphisms.
Before stating the axiom, we illustrate it in our two motivating examples of $n$-categories.
@@ -506,7 +510,7 @@
\begin{axiom}[Product (identity) morphisms]
\label{axiom:product}
For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$),
-there is a map $\pi^*:\cC(X)\to \cC(E)$.
+there is a map $\pi^*:\cC(X)\trans{}\to \cC(E)$.
These maps must satisfy the following conditions.
\begin{enumerate}
\item
@@ -529,7 +533,7 @@
but $X_1 \cap X_2$ might not be codimension 1, and indeed we might have $X_1 = X_2 = X$.
We assume that there is a decomposition of $X$ into balls which is compatible with
$X_1$ and $X_2$.
-Let $a\in \cC(X)$, and let $a_i$ denote the restriction of $a$ to $X_i\sub X$.
+Let $a\in \cC(X)\trans{}$, and let $a_i$ denote the restriction of $a$ to $X_i\sub X$.
(We assume that $a$ is splittable with respect to the above decomposition of $X$ into balls.)
Then
\[