# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1301494574 25200
# Node ID 6fd9b377be3be5b8c1c2743ef1542879bbc1c23d
# Parent  ae93002b511e7570f82e0c7d0403ebd26f22d5ad
fix definition of refinement of ball decomp (intermediate manifolds are disj unions of balls)

diff -r ae93002b511e -r 6fd9b377be3b text/ncat.tex
--- a/text/ncat.tex	Thu Mar 24 10:06:09 2011 -0700
+++ b/text/ncat.tex	Wed Mar 30 07:16:14 2011 -0700
@@ -987,8 +987,11 @@
 and we  will define $\cl{\cC}(W)$ as a suitable colimit 
 (or homotopy colimit in the $A_\infty$ case) of this functor. 
 We'll later give a more explicit description of this colimit.
-In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-balls with boundary data), 
-then the resulting colimit is also enriched, that is, the set associated to $W$ splits into subsets according to boundary data, and each of these subsets has the appropriate structure (e.g. a vector space or chain complex).
+In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain 
+complexes to $n$-balls with boundary data), 
+then the resulting colimit is also enriched, that is, the set associated to $W$ splits into 
+subsets according to boundary data, and each of these subsets has the appropriate structure 
+(e.g. a vector space or chain complex).
 
 Recall (Definition \ref{defn:gluing-decomposition}) that a {\it ball decomposition} of $W$ is a 
 sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls
@@ -1005,7 +1008,8 @@
 
 Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement
 of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$
-with $\du_b Y_b = M_i$ for some $i$.
+with $\du_b Y_b = M_i$ for some $i$,
+and with $M_0,\ldots, M_i$ each being a disjoint union of balls.
 
 \begin{defn}
 The poset $\cell(W)$ has objects the permissible decompositions of $W$, 
@@ -1036,7 +1040,7 @@
 	\psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl
 \end{equation}
 where the restrictions to the various pieces of shared boundaries amongst the cells
-$X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells).
+$X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n{-}1$-cells).
 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$.
 \end{defn}