# HG changeset patch
# User Scott Morrison <scott@tqft.net>
# Date 1288038972 25200
# Node ID 8378e03d3c7fdf8513b3e07bce9229b805a00a70
# Parent  e0f5ec5827254a6fb45b02f7dcfe2270269dd2c5
starting on cell decompositions

diff -r e0f5ec582725 -r 8378e03d3c7f pnas/diagrams/ncat/zz2.pdf
Binary file pnas/diagrams/ncat/zz2.pdf has changed
diff -r e0f5ec582725 -r 8378e03d3c7f pnas/pnas.tex
--- a/pnas/pnas.tex	Mon Oct 25 13:08:15 2010 -0700
+++ b/pnas/pnas.tex	Mon Oct 25 13:36:12 2010 -0700
@@ -186,6 +186,51 @@
 }
 \subsection{The blob complex}
 \subsubsection{Decompositions of manifolds}
+
+A {\emph 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
+$\du_a X_a$.
+
+If $X_a$ is some component of $M_0$, note that its image in $W$ need not be a ball; parts of $\bd X_a$ may have been glued together.
+Define a {\it permissible decomposition} of $W$ to be a map
+\[
+	\coprod_a X_a \to W,
+\]
+which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$.
+Roughly, a permissible decomposition is like a ball decomposition where we don't care in which order the balls
+are glued up to yield $W$, so long as there is some (non-pathological) way to glue them.
+
+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 $i$.
+
+\begin{defn}
+The poset $\cell(W)$ has objects the permissible decompositions of $W$, 
+and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.
+See Figure \ref{partofJfig} for an example.
+\end{defn}
+
+
+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,
+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.
+For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset
+\begin{equation}
+\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 $\cC$.
+\end{defn}
+
+
 \nn{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls}
 \subsubsection{Homotopy colimits}
 \nn{How can we extend an $n$-category from balls to arbitrary manifolds?}
@@ -473,6 +518,16 @@
 %% \caption{Almost Sharp Front}\label{afoto}
 %% \end{figure}
 
+
+\begin{figure}
+\begin{equation*}
+\mathfig{.23}{ncat/zz2}
+\end{equation*}
+\caption{A small part of $\cell(W)$}
+\label{partofJfig}
+\end{figure}
+
+
 %% For Tables, put caption above table
 %%
 %% Table caption should start with a capital letter, continue with lower case
diff -r e0f5ec582725 -r 8378e03d3c7f pnas/preamble.tex
--- a/pnas/preamble.tex	Mon Oct 25 13:08:15 2010 -0700
+++ b/pnas/preamble.tex	Mon Oct 25 13:36:12 2010 -0700
@@ -48,6 +48,7 @@
 \newtheorem{property}{Property}
 \newtheorem{prop}{Proposition}
 \newtheorem{thm}[prop]{Theorem}
+\newtheorem{defn}[prop]{Definition}
 
 \newenvironment{rem}{\noindent\textsl{Remark.}}{}
 
@@ -75,6 +76,12 @@
 \newcommand{\googlebooks}[1]{(preview at \href{http://books.google.com/books?id=#1}{google books})}
 
 
+% figures
+
+\newcommand{\mathfig}[2]{\ensuremath{\hspace{-3pt}\begin{array}{c}%
+  \raisebox{-2.5pt}{\includegraphics[width=#1\textwidth]{diagrams/#2}}%
+\end{array}\hspace{-3pt}}}
+
 
 % packages