# HG changeset patch
# User kevin@6e1638ff-ae45-0410-89bd-df963105f760
# Date 1256184529 0
# Node ID f8d909559d19791ea7c8eb48300836a4de2083fc
# Parent  7b4f5e36d9deba70dde3252ef963314983a42a13
...

diff -r 7b4f5e36d9de -r f8d909559d19 blob1.tex
--- a/blob1.tex	Tue Oct 20 18:25:54 2009 +0000
+++ b/blob1.tex	Thu Oct 22 04:08:49 2009 +0000
@@ -6,7 +6,8 @@
 \input{text/kw_macros}
 
 
-\title{Blob Homology}
+%\title{Blob Homology}
+\title{The Blob Complex}
 
 \begin{document}
 
@@ -20,11 +21,10 @@
 \textbf{Draft version, do not distribute.}
 
 %\versioninfo
-[19 October 2009, rev. 129]
+%[19 October 2009, rev. $>$ 129]
 
 \medskip
 
-
 \noindent
 {\bf Warning:} This draft is draftier than you might expect.
 More specifically,
@@ -45,22 +45,26 @@
 On the other hand, if you are only going to read this paper once,
 {\bf then don't read this version,} as a more complete version will be available in a couple of months.
 
-\noop{
+\tableofcontents
 
-\section*{Todo}
+%\noop{
+
+\section*{Old To Do list}
 
 \subsection*{What else?...}
 
+\relax
+
+\nn{should move this to do list elsewhere}
+
 \begin{itemize}
 \item higher priority
 \begin{itemize}
 \item K\&S: learn the state of the art in A-inf categories
 (tensor products, Kadeishvili result, ...)
-\item K: so-called evaluation map stuff
-\item K: topological fields
 \item section describing intended applications
 \item say something about starting with semisimple n-cat (trivial?? not trivial?)
-\item T.O.C.
+\item need to make up our minds about quasi-isomorphic vs hty equivalent
 \end{itemize}
 \item medium priority
 \begin{itemize}
@@ -68,7 +72,6 @@
 \item dimension $n+1$ (generalized Deligne conjecture?)
 \item should be clear about PL vs Diff; probably PL is better
 (or maybe not)
-\item something about higher derived coend things (derived 2-coend, e.g.)
 \item shuffle product vs gluing product (?)
 \item connection between $A_\infty$ operad and topological $A_\infty$ cat defs
 \end{itemize}
@@ -80,7 +83,7 @@
 \end{itemize}
 \end{itemize}
 
-} %end \noop
+%} %end \noop
 
 
 
diff -r 7b4f5e36d9de -r f8d909559d19 text/intro.tex
--- a/text/intro.tex	Tue Oct 20 18:25:54 2009 +0000
+++ b/text/intro.tex	Thu Oct 22 04:08:49 2009 +0000
@@ -2,7 +2,30 @@
 
 \section{Introduction}
 
-[Outline for intro]
+[some things to cover in the intro]
+\begin{itemize}
+\item explain relation between old and new blob complex definitions
+\item overview of sections
+\item state main properties of blob complex (already mostly done below)
+\item give multiple motivations/viewpoints for blob complex: (1) derived cat
+version of TQFT Hilbert space; (2) generalization of Hochschild homology to higher $n$-cats;
+(3) ? sort-of-obvious colimit type construction;
+(4) ? a generalization of $C_*(\Maps(M, T))$ to the case where $T$ is
+a category rather than a manifold
+\item hope to apply to Kh, contact, (other examples?) in the future
+\item ?? we have resisted the temptation 
+(actually, it was not a temptation) to state things in the greatest
+generality possible
+\item related: we are being unsophisticated from a homotopy theory point of
+view and using chain complexes in many places where we could be by with spaces
+\item ? one of the points we make (far) below is that there is not really much
+difference between (a) systems of fields and local relations and (b) $n$-cats;
+thus we tend to switch between talking in terms of one or the other
+\end{itemize}
+
+\medskip\hrule\medskip
+
+[Old outline for intro]
 \begin{itemize}
 \item Starting point: TQFTs via fields and local relations.
 This gives a satisfactory treatment for semisimple TQFTs
@@ -61,18 +84,20 @@
 \hrule
 \bigskip
 
-We then show that blob homology enjoys the following
-\ref{property:gluing} properties.
+We then show that blob homology enjoys the following properties.
 
 \begin{property}[Functoriality]
 \label{property:functoriality}%
-Blob homology is functorial with respect to diffeomorphisms. That is, fixing an $n$-dimensional system of fields $\cF$ and local relations $\cU$, the association
+Blob homology is functorial with respect to homeomorphisms. That is, 
+for fixed $n$-category / fields $\cC$, the association
 \begin{equation*}
-X \mapsto \bc_*^{\cF,\cU}(X)
+X \mapsto \bc_*^{\cC}(X)
 \end{equation*}
-is a functor from $n$-manifolds and diffeomorphisms between them to chain complexes and isomorphisms between them.
+is a functor from $n$-manifolds and homeomorphisms between them to chain complexes and isomorphisms between them.
 \end{property}
 
+\nn{should probably also say something about being functorial in $\cC$}
+
 \begin{property}[Disjoint union]
 \label{property:disjoint-union}
 The blob complex of a disjoint union is naturally the tensor product of the blob complexes.
@@ -81,13 +106,19 @@
 \end{equation*}
 \end{property}
 
-\begin{property}[A map for gluing]
+\begin{property}[Gluing map]
 \label{property:gluing-map}%
 If $X_1$ and $X_2$ are $n$-manifolds, with $Y$ a codimension $0$-submanifold of $\bdy X_1$, and $Y^{\text{op}}$ a codimension $0$-submanifold of $\bdy X_2$,
 there is a chain map
 \begin{equation*}
 \gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2).
 \end{equation*}
+\nn{alternate version:}Given a gluing $X_\mathrm{cut} \to X_\mathrm{gl}$, there is
+a natural map
+\[
+	\bc_*(X_\mathrm{cut}) \to \bc_*(X_\mathrm{gl}) .
+\]
+(Natural with respect to homeomorphisms, and also associative with respect to iterated gluings.)
 \end{property}
 
 \begin{property}[Contractibility]
@@ -104,9 +135,9 @@
 \label{property:skein-modules}%
 The $0$-th blob homology of $X$ is the usual 
 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$
-by $(\cF,\cU)$. (See \S \ref{sec:local-relations}.)
+by $\cC$. (See \S \ref{sec:local-relations}.)
 \begin{equation*}
-H_0(\bc_*^{\cF,\cU}(X)) \iso A^{\cF,\cU}(X)
+H_0(\bc_*^{\cC}(X)) \iso A^{\cC}(X)
 \end{equation*}
 \end{property}