# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1308526294 21600
# Node ID e3ddb8605e3296ebdb074bb62219191ce6c234df
# Parent  33b3e0c065d279f2b07de84bd3afe7a3e8383350
adding transversality requirement to product morphism axiom

diff -r 33b3e0c065d2 -r e3ddb8605e32 blob to-do
--- 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
diff -r 33b3e0c065d2 -r e3ddb8605e32 blob_changes_v3
--- 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
 
 
 
diff -r 33b3e0c065d2 -r e3ddb8605e32 text/ncat.tex
--- 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 
 \[