--- 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
--- 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
+
--- 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 $\psi_{\cC;W}(x)$.