# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1305261754 25200
# Node ID b76b4b79dbe1c808c83ff2c5ef856fd6e8e69341
# Parent  c24e59300fcaf9085b5103bcdc255749cd876b9d
starting to work on colimit stuff, but not much progress yet

diff -r c24e59300fca -r b76b4b79dbe1 blob to-do
--- a/blob to-do	Tue May 10 14:30:23 2011 -0700
+++ b/blob to-do	Thu May 12 21:42:34 2011 -0700
@@ -48,9 +48,6 @@
 
 modules:
 
-* Marked hemispheres, need better language.
-	- add something like "The is just a ball (\bd N \ N) with its entire boundary (\nd N) marked.  We use the term hemisphere because these balls are half the boundary of a larger ball.
-
 * Lemma 6.4.5 needs to actually construct this map! Needs more input! Do
 we actually need this as written?
 	- KW will look at it; probably needs to be weakened
@@ -70,3 +67,5 @@
 
 * lemma [inject  6.3.5?] assumes more splittablity than the axioms imply (?)
 
+* figure for example 3.1.2 (sin 1/z)
+
diff -r c24e59300fca -r b76b4b79dbe1 text/blobdef.tex
--- a/text/blobdef.tex	Tue May 10 14:30:23 2011 -0700
+++ b/text/blobdef.tex	Thu May 12 21:42:34 2011 -0700
@@ -158,7 +158,7 @@
 a manifold.
 Thus will need to be more careful when speaking of a field $r$ on the complement of the blobs.
 
-\begin{example}
+\begin{example} \label{sin1x-example}
 Consider the four subsets of $\Real^3$,
 \begin{align*}
 A & = [0,1] \times [0,1] \times [0,1] \\
diff -r c24e59300fca -r b76b4b79dbe1 text/ncat.tex
--- a/text/ncat.tex	Tue May 10 14:30:23 2011 -0700
+++ b/text/ncat.tex	Thu May 12 21:42:34 2011 -0700
@@ -1037,7 +1037,7 @@
 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 $i$,
-and with $M_0,\ldots, M_i$ each being a disjoint union of balls.
+and with $M_0, M_1, \ldots, M_i$ each being a disjoint union of balls.
 
 \begin{defn}
 The poset $\cell(W)$ has objects the permissible decompositions of $W$, 
@@ -1056,20 +1056,31 @@
 An $n$-category $\cC$ determines 
 a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets 
 (possibly with additional structure if $k=n$).
-Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls,
+For pedagogical reasons, let us first the case where a decomposition $y$ of $W$ is a nice, non-pathological
+cell decomposition.
+Then each $k$-ball $X$ of $y$ has its boundary decomposed into $k{-}1$-balls,
 and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries
 are splittable along this decomposition.
 
-\begin{defn}
-Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows.
+We can now
+define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows.
 For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset
 \begin{equation}
-\label{eq:psi-C}
+%\label{eq:psi-C}
 	\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).
 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 $\cC$.
+
+In general, $y$ might be more general than a cell decomposition
+(see Example \ref{sin1x-example}), so we must define $\psi_{\cC;W}$ in a more roundabout way.
+\nn{...}
+
+\begin{defn}
+Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows.
+\nn{...}
+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}
 
 If $k=n$ in the above definition and we are enriching in some auxiliary category,