--- a/sandbox.tex Wed Aug 25 22:58:41 2010 -0700
+++ b/sandbox.tex Thu Aug 26 13:20:13 2010 -0700
@@ -12,9 +12,70 @@
\begin{document}
-\begin{equation*}
-\mathfig{.73}{tempkw/zo2}
-\end{equation*}
+
+\begin{figure}[t]
+\begin{center}
+\begin{tikzpicture}
+
+\node(A) at (-4,0) {
+\begin{tikzpicture}[scale=.8, fill=blue!15!white]
+\filldraw[line width=1.5pt] (-.4,1) .. controls +(-1,-.1) and +(-1,0) .. (0,-1)
+ .. controls +(1,0) and +(1,-.1) .. (.4,1) -- (.4,3)
+ .. controls +(3,-.4) and +(3,0) .. (0,-3)
+ .. controls +(-3,0) and +(-3,-.1) .. (-.4,3) -- cycle;
+\node at (0,-2) {$X$};
+\node (W) at (-2.7,-2) {$W$};
+\node (Y1) at (-1.2,3.5) {$Y$};
+\node (Y2) at (1.4,3.5) {$Y$};
+\node[outer sep=2.3] (y1e) at (-.4,2) {};
+\node[outer sep=2.3] (y2e) at (.4,2) {};
+\node (we1) at (-2.2,-1.1) {};
+\node (we2) at (-.6,-.7) {};
+\draw[->] (Y1) -- (y1e);
+\draw[->] (Y2) -- (y2e);
+\draw[->] (W) .. controls +(0,.5) and +(-.5,-.2) .. (we1);
+\draw[->] (W) .. controls +(.5,0) and +(-.2,-.5) .. (we2);
+\end{tikzpicture}
+};
+
+\node(B) at (4,0) {
+\begin{tikzpicture}[scale=.8, fill=blue!15!white]
+\fill (0,1) .. controls +(-1,0) and +(-1,0) .. (0,-1)
+ .. controls +(1,0) and +(1,0) .. (0,1) -- (0,3)
+ .. controls +(3,0) and +(3,0) .. (0,-3)
+ .. controls +(-3,0) and +(-3,0) .. (0,3) -- cycle;
+\draw[line width=1.5pt] (0,1) .. controls +(-1,0) and +(-1,0) .. (0,-1)
+ .. controls +(1,0) and +(1,0) .. (0,1);
+\draw[line width=1.5pt] (0,3) .. controls +(3,0) and +(3,0) .. (0,-3)
+ .. controls +(-3,0) and +(-3,0) .. (0,3);
+\draw[line width=.5pt, black!65!white] (0,1) -- (0,3);
+\node at (0,-2) {$X\sgl$};
+\node (W) at (2.7,-2) {$W\sgl$};
+\node (we1) at (2.2,-1.1) {};
+\node (we2) at (.6,-.7) {};
+\draw[->] (W) .. controls +(0,.5) and +(.5,-.2) .. (we1);
+\draw[->] (W) .. controls +(-.5,0) and +(.2,-.5) .. (we2);
+\end{tikzpicture}
+};
+
+
+\draw[->, red!80!green, line width=2pt] (A) -- node[above, black] {glue} (B);
+
+\end{tikzpicture}
+\end{center}
+\caption{Gluing with corners}
+\label{fig:gluing-with-corners}
+\end{figure}
+
+
+
+
+
+blah
+
+\vfill\eject
+
+
\begin{tikzpicture}
\newcommand{\rr}{6}
--- a/text/intro.tex Wed Aug 25 22:58:41 2010 -0700
+++ b/text/intro.tex Thu Aug 26 13:20:13 2010 -0700
@@ -35,14 +35,20 @@
We expect applications of the blob complex to contact topology and Khovanov homology
but do not address these in this paper.
-See \S \ref{sec:future} for slightly more detail.
+%See \S \ref{sec:future} for slightly more detail.
+
+Throughout, we have resisted the temptation to work in the greatest possible generality.
+(Don't worry, it wasn't that hard.)
+In most of the places where we say ``set" or ``vector space", any symmetric monoidal category
+with sufficient limits and colimits would do.
+We could also replace many of our chain complexes with topological spaces (or indeed, work at the generality of model categories).
\subsection{Structure of the paper}
The subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}),
-summarize the formal properties of the blob complex (see \S \ref{sec:properties}), describe known specializations (see \S \ref{sec:specializations}), outline the major results of the paper (see \S \ref{sec:structure} and \S \ref{sec:applications})
-and outline anticipated future directions (see \S \ref{sec:future}).
-\nn{recheck this list after done editing intro}
+summarize the formal properties of the blob complex (see \S \ref{sec:properties}), describe known specializations (see \S \ref{sec:specializations}), and outline the major results of the paper (see \S \ref{sec:structure} and \S \ref{sec:applications}).
+%and outline anticipated future directions (see \S \ref{sec:future}).
+%\nn{recheck this list after done editing intro}
The first part of the paper (sections \S \ref{sec:fields}---\S \ref{sec:evaluation}) gives the definition of the blob complex,
and establishes some of its properties.
@@ -455,7 +461,7 @@
-
+\noop{
\subsection{Future directions}
\label{sec:future}
\nn{KW: Perhaps we should delete this subsection and salvage only the first few sentences.}
@@ -482,7 +488,7 @@
but haven't investigated the details.
Most importantly, however, \nn{applications!} \nn{cyclic homology, $n=2$ cases, contact, Kh} \nn{stabilization} \nn{stable categories, generalized cohomology theories}
-
+} %%% end \noop
\subsection{Thanks and acknowledgements}
% attempting to make this chronological rather than alphabetical
@@ -499,5 +505,7 @@
and
Alexander Kirillov
for many interesting and useful conversations.
-During this work, Kevin Walker has been at Microsoft Station Q, and Scott Morrison has been at Microsoft Station Q and the Miller Institute for Basic Research at UC Berkeley. We'd like to thank the Aspen Center for Physics for the conducive environment provided there during the final preparation of this manuscript.
+During this work, Kevin Walker has been at Microsoft Station Q, and Scott Morrison has been at Microsoft Station Q and the Miller Institute for Basic Research at UC Berkeley. We'd like to thank the Aspen Center for Physics for the pleasant and productive
+% "conducive" needs an object; "conducive to blah"
+environment provided there during the final preparation of this manuscript.
--- a/text/ncat.tex Wed Aug 25 22:58:41 2010 -0700
+++ b/text/ncat.tex Thu Aug 26 13:20:13 2010 -0700
@@ -127,8 +127,9 @@
Most of the examples of $n$-categories we are interested in are enriched in the following sense.
The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and
all $c\in \cl{\cC}(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category
+with sufficient limits and colimits
(e.g.\ vector spaces, or modules over some ring, or chain complexes),
-\nn{actually, need both disj-union/sub and product/tensor-product; what's the name for this sort of cat?}
+%\nn{actually, need both disj-union/sum and product/tensor-product; what's the name for this sort of cat?}
and all the structure maps of the $n$-category should be compatible with the auxiliary
category structure.
Note that this auxiliary structure is only in dimension $n$; if $\dim(Y) < n$ then
--- a/text/tqftreview.tex Wed Aug 25 22:58:41 2010 -0700
+++ b/text/tqftreview.tex Thu Aug 26 13:20:13 2010 -0700
@@ -115,7 +115,60 @@
Let $\bd X = (Y \du Y) \cup W$, where the two copies of $Y$
are disjoint from each other and $\bd(Y\du Y) = \bd W$.
Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$
-(Figure \ref{fig:???}).
+(Figure \ref{fig:gluing-with-corners}).
+\begin{figure}[t]
+\begin{center}
+\begin{tikzpicture}
+
+\node(A) at (-4,0) {
+\begin{tikzpicture}[scale=.8, fill=blue!15!white]
+\filldraw[line width=1.5pt] (-.4,1) .. controls +(-1,-.1) and +(-1,0) .. (0,-1)
+ .. controls +(1,0) and +(1,-.1) .. (.4,1) -- (.4,3)
+ .. controls +(3,-.4) and +(3,0) .. (0,-3)
+ .. controls +(-3,0) and +(-3,-.1) .. (-.4,3) -- cycle;
+\node at (0,-2) {$X$};
+\node (W) at (-2.7,-2) {$W$};
+\node (Y1) at (-1.2,3.5) {$Y$};
+\node (Y2) at (1.4,3.5) {$Y$};
+\node[outer sep=2.3] (y1e) at (-.4,2) {};
+\node[outer sep=2.3] (y2e) at (.4,2) {};
+\node (we1) at (-2.2,-1.1) {};
+\node (we2) at (-.6,-.7) {};
+\draw[->] (Y1) -- (y1e);
+\draw[->] (Y2) -- (y2e);
+\draw[->] (W) .. controls +(0,.5) and +(-.5,-.2) .. (we1);
+\draw[->] (W) .. controls +(.5,0) and +(-.2,-.5) .. (we2);
+\end{tikzpicture}
+};
+
+\node(B) at (4,0) {
+\begin{tikzpicture}[scale=.8, fill=blue!15!white]
+\fill (0,1) .. controls +(-1,0) and +(-1,0) .. (0,-1)
+ .. controls +(1,0) and +(1,0) .. (0,1) -- (0,3)
+ .. controls +(3,0) and +(3,0) .. (0,-3)
+ .. controls +(-3,0) and +(-3,0) .. (0,3) -- cycle;
+\draw[line width=1.5pt] (0,1) .. controls +(-1,0) and +(-1,0) .. (0,-1)
+ .. controls +(1,0) and +(1,0) .. (0,1);
+\draw[line width=1.5pt] (0,3) .. controls +(3,0) and +(3,0) .. (0,-3)
+ .. controls +(-3,0) and +(-3,0) .. (0,3);
+\draw[line width=.5pt, black!65!white] (0,1) -- (0,3);
+\node at (0,-2) {$X\sgl$};
+\node (W) at (2.7,-2) {$W\sgl$};
+\node (we1) at (2.2,-1.1) {};
+\node (we2) at (.6,-.7) {};
+\draw[->] (W) .. controls +(0,.5) and +(.5,-.2) .. (we1);
+\draw[->] (W) .. controls +(-.5,0) and +(.2,-.5) .. (we2);
+\end{tikzpicture}
+};
+
+
+\draw[->, red!50!green, line width=2pt] (A) -- node[above, black] {glue} (B);
+
+\end{tikzpicture}
+\end{center}
+\caption{Gluing with corners}
+\label{fig:gluing-with-corners}
+\end{figure}
Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself
(without corners) along two copies of $\bd Y$.
Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a splittable field on $W\sgl$ and let