# HG changeset patch
# User kevin@6e1638ff-ae45-0410-89bd-df963105f760
# Date 1250549468 0
# Node ID 638be64bd329d1e10a73f00cbe7dcc0f2f66650e
# Parent  0df8bde1c8966db540f5aa50eb34cfda893cc242
...

diff -r 0df8bde1c896 -r 638be64bd329 blob1.tex
--- a/blob1.tex	Mon Aug 17 05:23:35 2009 +0000
+++ b/blob1.tex	Mon Aug 17 22:51:08 2009 +0000
@@ -73,6 +73,8 @@
 
 \input{text/ncat}
 
+\input{text/a_inf_blob}
+
 \input{text/A-infty}
 
 \input{text/gluing}
diff -r 0df8bde1c896 -r 638be64bd329 text/A-infty.tex
--- a/text/A-infty.tex	Mon Aug 17 05:23:35 2009 +0000
+++ b/text/A-infty.tex	Mon Aug 17 22:51:08 2009 +0000
@@ -3,6 +3,9 @@
 \section{Homological systems of fields}
 \label{sec:homological-fields}
 
+\nn{*** If we keep Section \ref{sec:ncats}, then this section becomes obsolete.
+Retain it for now.}
+
 In this section, we extend the definition of blob homology to allow \emph{homological systems of fields}.
 
 We begin with a definition of a \emph{topological $A_\infty$ category}, and then introduce the notion of a homological system of fields. A topological $A_\infty$ category gives a $1$-dimensional homological system of fields. We'll suggest that any good definition of a topological $A_\infty$ $n$-category with duals should allow construction of an $n$-dimensional homological system of fields, but we won't propose any such definition here. Later, we extend the definition of blob homology to allow homological fields as input. These definitions allow us to state and prove a theorem about the blob homology of a product manifold, and an intermediate theorem about gluing, in preparation for the proof of Property \ref{property:gluing}.
diff -r 0df8bde1c896 -r 638be64bd329 text/a_inf_blob.tex
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/text/a_inf_blob.tex	Mon Aug 17 22:51:08 2009 +0000
@@ -0,0 +1,60 @@
+%!TEX root = ../blob1.tex
+
+\section{The blob complex for $A_\infty$ $n$-categories}
+\label{sec:ainfblob}
+
+Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we define the blob
+complex $\bc_*(M)$ to the be the colimit $\cC(M)$ of Section \ref{sec:ncats}.
+\nn{say something about this being anticlimatically tautological?}
+We will show below 
+\nn{give ref}
+that this agrees (up to homotopy) with our original definition of the blob complex
+in the case of plain $n$-categories.
+When we need to distinguish between the new and old definitions, we will refer to the 
+new-fangled and old-fashioned blob complex.
+
+\medskip
+
+Let $M^n = Y^k\times F^{n-k}$.  
+Let $C$ be a plain $n$-category.
+Let $\cF$ be the $A_\infty$ $k$-category which assigns to a $k$-ball
+$X$ the old-fashioned blob complex $\bc_*(X\times F)$.
+
+\begin{thm}
+The old-fashioned blob complex $\bc_*^C(Y\times F)$ is homotopy equivalent to the
+new-fangled blob complex $\bc_*^\cF(Y)$.
+\end{thm}
+
+\begin{proof}
+We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}.
+
+First we define a map from $\bc_*^\cF(Y)$ to $\bc_*^C(Y\times F)$.
+In filtration degree 0 we just glue together the various blob diagrams on $X\times F$
+(where $X$ is a component of a permissible decomposition of $Y$) to get a blob diagram on
+$Y\times F$.
+In filtration degrees 1 and higher we define the map to be zero.
+It is easy to check that this is a chain map.
+
+Next we define a map from $\bc_*^C(Y\times F)$ to $\bc_*^\cF(Y)$.
+Actually, we will define it on the homotopy equivalent subcomplex
+$\cS_* \sub \bc_*^C(Y\times F)$ generated by blob diagrams which are small with respect to some open cover
+of $Y\times F$.
+\nn{need reference to small blob lemma}
+We will have to show eventually that this is independent (up to homotopy) of the choice of cover.
+Also, for a fixed choice of cover we will only be able to define the map for blob degree less than
+some bound, but this bound goes to infinity as the cover become finer.
+
+\nn{....}
+\end{proof}
+
+\nn{need to say something about dim $< n$ above}
+
+
+
+\medskip
+\hrule
+\medskip
+
+\nn{to be continued...}
+\medskip
+
diff -r 0df8bde1c896 -r 638be64bd329 text/gluing.tex
--- a/text/gluing.tex	Mon Aug 17 05:23:35 2009 +0000
+++ b/text/gluing.tex	Mon Aug 17 22:51:08 2009 +0000
@@ -3,6 +3,8 @@
 \section{Gluing - needs to be rewritten/replaced}
 \label{sec:gluing}%
 
+\nn{*** this section is now obsolete; should be removed soon}
+
 We now turn to establishing the gluing formula for blob homology, restated from Property \ref{property:gluing} in the Introduction
 \begin{itemize}
 %\mbox{}% <-- gets the indenting right
diff -r 0df8bde1c896 -r 638be64bd329 text/ncat.tex
--- a/text/ncat.tex	Mon Aug 17 05:23:35 2009 +0000
+++ b/text/ncat.tex	Mon Aug 17 22:51:08 2009 +0000
@@ -410,6 +410,7 @@
 
 
 \subsection{From $n$-categories to systems of fields}
+\label{ss:ncat_fields}
 
 We can extend the functors $\cC$ above from $k$-balls to arbitrary $k$-manifolds as follows.
 
@@ -466,13 +467,13 @@
 
 In the $A_\infty$ case enriched over chain complexes, the concrete description of the colimit
 is as follows.
-Define an $m$-sequence to be a sequence $x_0 \le x_1 \le \dots \le x_{m-1}$ of permissible decompositions.
+Define an $m$-sequence to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions.
 Such sequences (for all $m$) form a simplicial set.
 Let
 \[
 	V = \bigoplus_{(x_i)} \psi_\cC(x_0) ,
 \]
-where the sum is over all $m$-sequences and all $m$.
+where the sum is over all $m$-sequences and all $m$, and each summand is degree shifted by $m$.
 We endow $V$ with a differential which is the sum of the differential of the $\psi_\cC(x_0)$
 summands plus another term using the differential of the simplicial set of $m$-sequences.
 More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$
@@ -487,6 +488,14 @@
 combine only two balls at a time; for $n=1$ this version will lead to usual definition
 of $A_\infty$ category}
 
+We will call $m$ the filtration degree of the complex.
+We can think of this construction as starting with a disjoint copy of a complex for each
+permissible decomposition (filtration degree 0).
+Then we glue these together with mapping cylinders coming from gluing maps
+(filtration degree 1).
+Then we kill the extra homology we just introduced with mapping cylinder between the mapping cylinders (filtration degree 2).
+And so on.
+
 $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds.
 
 It is easy to see that