
diff -r eb9de49b98b4 -r d4e6bf589ebe text/ncat.tex
--- a/text/ncat.tex	Wed Oct 07 18:33:41 2009 +0000
+++ b/text/ncat.tex	Tue Oct 13 21:32:06 2009 +0000
@@ -218,7 +218,7 @@
 \[
 	(a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') .
 \]
-\nn{problem: if pinched boundary, then only one factor}
+\nn{if pinched boundary, then remove first case above}
 Product morphisms are associative:
 \[
 	(a\times D)\times D' = a\times (D\times D') .
@@ -234,6 +234,14 @@
 
 \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 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.}
+
 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.
@@ -260,6 +268,8 @@
 Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball.
 Let $J$ be a 1-ball (interval).
 We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$.
+(Here we use the ``pinched" version of $Y\times J$.
+\nn{need notation for this})
 We define a map
 \begin{eqnarray*}
 	\psi_{Y,J}: \cC(X) &\to& \cC(X) \\
@@ -873,7 +883,7 @@
 \item say something about unoriented vs oriented vs spin vs pin for $n=1$ (and $n=2$?)
 \item spell out what difference (if any) Top vs PL vs Smooth makes
 \item explain relation between old-fashioned blob homology and new-fangled blob homology
-(follows as special case of product formula (product with a point).
+(follows as special case of product formula (product with a point)).
 \item define $n{+}1$-cat of $n$-cats (a.k.a.\ $n{+}1$-category of generalized bimodules
 a.k.a.\ $n{+}1$-category of sphere modules); discuss Morita equivalence
 \end{itemize}




