# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1304870753 25200
# Node ID 775b5ca42bed40522e5b6829694a5391e93e259d
# Parent  73fc4868c0393c38ff2b1f76e1ba7474ef5416c3
make sure poset of decomps is a small category; added to to-do list

diff -r 73fc4868c039 -r 775b5ca42bed blob to-do
--- a/blob to-do	Sat May 07 09:40:20 2011 -0700
+++ b/blob to-do	Sun May 08 09:05:53 2011 -0700
@@ -42,8 +42,6 @@
 
 * Labeling of the k-1 skeleton agreeing on the k-2 skeleton is awfully vague. 
 
-* Category of permissible decompositions must be a small category in order to take the colimit. 
-
 * Boundary of \cl; not so easy to see!
 
 
@@ -62,3 +60,9 @@
 
 * SCOTT: make sure acknowledge list doesn't omit anyone from blob seminar (I think I have all the speakers)
 
+
+* review colors in figures
+
+* ? define Morita equivalence?
+
+* maybe put most figures at top of page
\ No newline at end of file
diff -r 73fc4868c039 -r 775b5ca42bed blob_changes_v3
--- a/blob_changes_v3	Sat May 07 09:40:20 2011 -0700
+++ b/blob_changes_v3	Sun May 08 09:05:53 2011 -0700
@@ -18,5 +18,6 @@
 - added remark about manifolds which do not admit ball decompositions; restricted product theorem (7.1.1) to apply only to these manifolds
 - added remarks about categories of defects
 - clarified that the "cell complexes" in string diagrams are actually a bit more general
-- 
+- added remark to insure that the poset of decompositions is a small category
 
+
diff -r 73fc4868c039 -r 775b5ca42bed text/ncat.tex
--- a/text/ncat.tex	Sat May 07 09:40:20 2011 -0700
+++ b/text/ncat.tex	Sun May 08 09:05:53 2011 -0700
@@ -1026,8 +1026,13 @@
 are glued up to yield $W$, so long as there is some (non-pathological) way to glue them.
 
 (Every smooth or PL manifold has a ball decomposition, but certain topological manifolds (e.g.\ non-smoothable
-topological 4-manifolds) do nat have ball decompositions.
-For such manifolds we have only the empty colimit.) 
+topological 4-manifolds) do not have ball decompositions.
+For such manifolds we have only the empty colimit.)
+
+We want the category (poset) of decompositions of $W$ to be small, so when we say decomposition we really
+mean isomorphism class of decomposition.
+Isomorphisms are defined in the obvious way: a collection of homeomorphisms $M_i\to M_i'$ which commute
+with the gluing maps $M_i\to M_{i+1}$ and $M'_i\to M'_{i+1}$.
 
 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 $\du_a X_a = M_0\to\cdots\to M_m = W$
@@ -1189,7 +1194,7 @@
 injective.
 Concretely, the colimit is the disjoint union of the sets (one for each decomposition of $W$),
 modulo the relation which identifies the domain of each of the injective maps
-with it's image.
+with its image.
 
 To save ink and electrons we will simplify notation and write $\psi(x)$ for $\psi_{\cC;W}(x)$.