--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/pnas/build.xml Wed Nov 17 11:26:00 2010 -0800
@@ -0,0 +1,82 @@
+<!-- This is an Ant build file; you'll need to install Ant before using it. -->
+<project name="pnas" default="usage">
+ <target name="init">
+ <property name="arxivTarFile" value="pnas.tar"/>
+ </target>
+
+ <!-- USAGE -->
+ <!-- Instructions for using the build file -->
+ <!-- =================================================================== -->
+ <target name = "usage" depends = "init">
+ <echo message = ""/>
+ <echo message = "blob build instructions"/>
+ <echo message = "-------------------------------------------------------------"/>
+ <echo message = ""/>
+ <echo message = " available targets are:"/>
+ <echo message = ""/>
+ <echo message = " arxiv --> builds blob.tar.gz, for submission to the arxiv"/>
+ <echo message = " pdf --> builds blob.pdf"/>
+ <echo message = ""/>
+ <echo message = "-------------------------------------------------------------"/>
+ <echo message = ""/>
+ </target>
+ <!-- =================================================================== -->
+
+ <target name="clean" depends="init">
+ <delete>
+ <fileset dir=".">
+ <include name="*.toc"/>
+ <include name="*.log"/>
+ <include name="*.aux"/>
+ <include name="*.blg"/>
+ <include name="*.xyc"/>
+ <include name="*.out"/>
+ </fileset>
+ </delete>
+ </target>
+
+ <target name="arxiv" depends="clean">
+ <delete file="${arxivTarFile}"/>
+ <delete file="${arxivTarFile}.gz"/>
+ <tar destfile="${arxivTarFile}" basedir="." includes="**"
+ excludes="*.synctex*,*.dvi,*.ps,pnas.pdf,*.png,${arxivTarFile},${arxivTarFile}.gz,sandbox.*,bibliography/**,papers/**,talks/**,diagrams/obsolete/**,diagrams/latex2pdf/**,text/obsolete/**,.hg/**"
+ />
+ <gzip src="${arxivTarFile}" destfile="${arxivTarFile}.gz"/>
+ <delete file="${arxivTarFile}"/>
+ </target>
+
+ <target name="bbl" depends="init">
+ <exec executable="pdflatex">
+ <arg value="pnas"/>
+ </exec>
+ <exec executable="bibtex">
+ <arg value="pnas"/>
+ </exec>
+ </target>
+
+ <target name="pdf" depends="bbl">
+ <exec executable="pdflatex">
+ <arg value="pnas"/>
+ </exec>
+ </target>
+
+ <target name="copy-pdf" depends="pdf">
+ <exec executable="svn" dir="../../../Sites/tqft.net/papers/">
+ <arg value="up"/>
+ <arg value="--accept"/>
+ <arg value="theirs-full"/>
+ </exec>
+ <copy file="pnas.pdf" tofile="../../../Sites/tqft.net/papers/blobs-pnas.pdf"/>
+ <exec executable="svn" dir="../../../Sites/tqft.net/papers/">
+ <arg value="commit"/>
+ <arg value="-m"/>
+ <arg value="pnas"/>
+ </exec>
+ <exec executable="ssh">
+ <arg value="scottmorrison@tqft.net"/>
+ <arg value="svn"/>
+ <arg value="up"/>
+ <arg value="tqft.net"/>
+ </exec>
+ </target>
+</project>
--- a/pnas/pnas.tex Wed Nov 17 11:16:39 2010 -0800
+++ b/pnas/pnas.tex Wed Nov 17 11:26:00 2010 -0800
@@ -96,7 +96,7 @@
%% For titles, only capitalize the first letter
%% \title{Almost sharp fronts for the surface quasi-geostrophic equation}
-\title{$n$-categories, colimits and the blob complex}
+\title{Higher categories, colimits and the blob complex}
%% Enter authors via the \author command.
@@ -158,47 +158,43 @@
%% \subsection{}
%% \subsubsection{}
-\dropcap{T}he aim of this paper is to describe a derived category version of TQFTs.
+\dropcap{T}he aim of this paper is to describe a derived category analogue of topological quantum field theories.
For our purposes, an $n{+}1$-dimensional TQFT is a locally defined system of
-invariants of manifolds of dimensions 0 through $n+1$.
-The TQFT invariant $A(Y)$ of a closed $k$-manifold $Y$ is a linear $(n{-}k)$-category.
+invariants of manifolds of dimensions 0 through $n+1$. In particular,
+the TQFT invariant $A(Y)$ of a closed $k$-manifold $Y$ is a linear $(n{-}k)$-category.
If $Y$ has boundary then $A(Y)$ is a collection of $(n{-}k)$-categories which afford
a representation of the $(n{-}k{+}1)$-category $A(\bd Y)$.
(See \cite{1009.5025} and \cite{kw:tqft};
for a more homotopy-theoretic point of view see \cite{0905.0465}.)
We now comment on some particular values of $k$ above.
-By convention, a linear 0-category is a vector space, and a representation
+A linear 0-category is a vector space, and a representation
of a vector space is an element of the dual space.
-So a TQFT assigns to each closed $n$-manifold $Y$ a vector space $A(Y)$,
+Thus a TQFT assigns to each closed $n$-manifold $Y$ a vector space $A(Y)$,
and to each $(n{+}1)$-manifold $W$ an element of $A(\bd W)^*$.
-In fact we will be mainly be interested in so-called $(n{+}\epsilon)$-dimensional
-TQFTs which have nothing to say about $(n{+}1)$-manifolds.
-For the remainder of this paper we assume this case.
+For the remainder of this paper we will in fact be interested in so-called $(n{+}\epsilon)$-dimensional
+TQFTs, which are slightly weaker structures and do not assign anything to general $(n{+}1)$-manifolds, but only to mapping cylinders.
When $k=n-1$ we have a linear 1-category $A(S)$ for each $(n{-}1)$-manifold $S$,
and a representation of $A(\bd Y)$ for each $n$-manifold $Y$.
-The gluing rule for the TQFT in dimension $n$ states that
+The TQFT gluing rule in dimension $n$ states that
$A(Y_1\cup_S Y_2) \cong A(Y_1) \ot_{A(S)} A(Y_2)$,
where $Y_1$ and $Y_2$ and $n$-manifolds with common boundary $S$.
When $k=0$ we have an $n$-category $A(pt)$.
-This can be thought of as the local part of the TQFT, and the full TQFT can be constructed of $A(pt)$
+This can be thought of as the local part of the TQFT, and the full TQFT can be reconstructed from $A(pt)$
via colimits (see below).
We call a TQFT semisimple if $A(S)$ is a semisimple 1-category for all $(n{-}1)$-manifolds $S$
and $A(Y)$ is a finite-dimensional vector space for all $n$-manifolds $Y$.
-Examples of semisimple TQFTS include Witten-Reshetikhin-Turaev theories,
+Examples of semisimple TQFTs include Witten-Reshetikhin-Turaev theories,
Turaev-Viro theories, and Dijkgraaf-Witten theories.
These can all be given satisfactory accounts in the framework outlined above.
-(The WRT invariants need to be reinterpreted as $3{+}1$-dimensional theories in order to be
-extended all the way down to 0-manifolds.)
+(The WRT invariants need to be reinterpreted as $3{+}1$-dimensional theories with only a weak dependence on interiors in order to be
+extended all the way down to dimension 0.)
For other non-semisimple TQFT-like invariants, however, the above framework seems to be inadequate.
-
-\nn{temp}
-
For example, the gluing rule for 3-manifolds in Ozsv\'{a}th-Szab\'{o}/Seiberg-Witten theory
involves a tensor product over an $A_\infty$ 1-category associated to 2-manifolds \cite{1003.0598,1005.1248}.
Long exact sequences are important computational tools in these theories,
@@ -539,9 +535,12 @@
\subsubsection{Colimits}
-\nn{Motivation: How can we extend an $n$-category from balls to arbitrary manifolds?}
+Our definition of an $n$-category is essentially a collection of functors defined on $k$-balls (and homeomorphisms) for $k \leq n$ satisfying certain axioms. It is natural to consider extending such functors to the larger categories of all $k$-manifolds (again, with homeomorphisms). In fact, the axioms stated above explictly require such an extension to $k$-spheres for $k<n$.
+
+The natural construction achieving this is the colimit.
+\nn{continue}
+
\nn{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls?}
-\nn{Explain codimension colimits here too}
We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$
with coefficients in the $n$-category $\cC$ as the homotopy colimit along $\cell(W)$