--- a/text/a_inf_blob.tex	Mon May 31 13:27:24 2010 -0700
+++ b/text/a_inf_blob.tex	Mon May 31 17:27:17 2010 -0700
@@ -20,45 +20,36 @@
 \input{text/smallblobs}
 
 \subsection{A product formula}
+\label{ss:product-formula}
+
+\noop{
+Let $Y$ be a $k$-manifold, $F$ be an $n{-}k$-manifold, and 
+\[
+	E = Y\times F .
+\]
+Let $\cC$ be an $n$-category.
+Let $\cF$ be the $k$-category of Example \ref{ex:blob-complexes-of-balls}, 
+\[
+	\cF(X) = \cC(X\times F)
+\]
+for $X$ an $m$-ball with $m\le k$.
+}
+
+\nn{need to settle on notation; proof and statement are inconsistent}
 
 \begin{thm} \label{product_thm}
 Given a topological $n$-category $C$ and a $n{-}k$-manifold $F$, recall from Example \ref{ex:blob-complexes-of-balls} that there is an  $A_\infty$ $k$-category $C^{\times F}$ defined by
 \begin{equation*}
 C^{\times F}(B) = \cB_*(B \times F, C).
 \end{equation*}
-Now, given a $k$-manifold $Y$, there is a homotopy equivalence between the `old-fashioned' blob complex for $Y \times F$ with coefficients in $C$ and the `new-fangled' (i.e. homotopy colimit) blob complex for $Y$ with coefficients in $C^{\times F}$:
+Now, given a $k$-manifold $Y$, there is a homotopy equivalence between the `old-fashioned' blob complex for $Y \times F$ with coefficients in $C$ and the `new-fangled' (i.e.\ homotopy colimit) blob complex for $Y$ with coefficients in $C^{\times F}$:
 \begin{align*}
 \cB_*(Y \times F, C) & \htpy \cB_*(Y, C^{\times F})
 \end{align*}
 \end{thm}
 
-\nn{To do: remark on the case of a nontrivial fiber bundle.  
-I can think of two approaches.
-In the first (slick but maybe a little too tautological), we generalize the 
-notion of an $n$-category to an $n$-category {\it over a space $B$}.
-(Should be able to find precedent for this in a paper of PT.
-This idea came up in a conversation with him, so maybe should site him.)
-In this generalization, we replace the categories of balls with the categories 
-of balls equipped with maps to $B$.
-A fiber bundle $F\to E\to B$ gives an example of such an $n$-category:
-assign to $p:D\to B$ the blob complex $\bc_*(p^*(E))$.
-We can do the colimit thing over $B$ with coefficients in a n-cat-over-B.
-The proof below works essentially unchanged in this case to show that the colimit is the blob complex of the total space $E$.
-}
 
-\nn{The second approach: Choose a decomposition $B = \cup X_i$
-such that the restriction of $E$ to $X_i$ is a product $F\times X_i$.
-Choose the product structure as well.
-To each codim-1 face $D_i\cap D_j$ we have a bimodule ($S^0$-module).
-And more generally to each codim-$j$ face we have an $S^{j-1}$-module.
-Decorate the decomposition with these modules and do the colimit.
-}
-
-\nn{There is a version of this last construction for arbitrary maps $E \to B$
-(not necessarily a fibration).}
-
-
-\begin{proof}[Proof of Theorem \ref{product_thm}]
+\begin{proof}%[Proof of Theorem \ref{product_thm}]
 We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}.
 
 First we define a map 
@@ -78,8 +69,8 @@
 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}
+of $Y\times F$
+(Proposition \ref{thm:small-blobs}).
 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.
@@ -96,12 +87,13 @@
 Let $a\in \cS_m$ be a blob diagram on $Y\times F$.
 For $m$ sufficiently small there exists a decomposition $K$ of $Y$ into $k$-balls such that the
 codimension 1 cells of $K\times F$ miss the blobs of $a$, and more generally such that $a$ is splittable along (the codimension-1 part of) $K\times F$.
-Let $D(a)$ denote the subcomplex of $\bc_*^\cF(Y)$ generated by all $(a, \bar{K})$
-such that each $K_i$ has the aforementioned splittable property
-(see Subsection \ref{ss:ncat_fields}).
-\nn{need to define $D(a)$ more clearly; also includes $(b_j, \bar{K})$ where
+Let $D(a)$ denote the subcomplex of $\bc_*^\cF(Y)$ generated by all $(a, \ol{K})$
+such that each $K_i$ has the aforementioned splittable property.
+(Recall that $\ol{K} = (K_0,\ldots,K_l)$ denotes a chain of decompositions;
+see Subsection \ref{ss:ncat_fields}.)
+\nn{need to define $D(a)$ more clearly; also includes $(b_j, \ol{K})$ where
 $\bd(a) = \sum b_j$.}
-(By $(a, \bar{K})$ we really mean $(a^\sharp, \bar{K})$, where $a^\sharp$ is 
+(By $(a, \ol{K})$ we really mean $(a^\sharp, \ol{K})$, where $a^\sharp$ is 
 $a$ split according to $K_0\times F$.
 To simplify notation we will just write plain $a$ instead of $a^\sharp$.)
 Roughly speaking, $D(a)$ consists of filtration degree 0 stuff which glues up to give
@@ -220,7 +212,8 @@
 
 We now show that $\phi\circ\psi$ and $\psi\circ\phi$ are homotopic to the identity.
 
-$\psi\circ\phi$ is the identity.  $\phi$ takes a blob diagram $a$ and chops it into pieces 
+$\psi\circ\phi$ is the identity on the nose.  
+$\phi$ takes a blob diagram $a$ and chops it into pieces 
 according to some decomposition $K$ of $Y$.
 $\psi$ glues those pieces back together, yielding the same $a$ we started with.
 
@@ -244,6 +237,33 @@
 
 \medskip
 
+\nn{To do: remark on the case of a nontrivial fiber bundle.  
+I can think of two approaches.
+In the first (slick but maybe a little too tautological), we generalize the 
+notion of an $n$-category to an $n$-category {\it over a space $B$}.
+(Should be able to find precedent for this in a paper of PT.
+This idea came up in a conversation with him, so maybe should site him.)
+In this generalization, we replace the categories of balls with the categories 
+of balls equipped with maps to $B$.
+A fiber bundle $F\to E\to B$ gives an example of such an $n$-category:
+assign to $p:D\to B$ the blob complex $\bc_*(p^*(E))$.
+We can do the colimit thing over $B$ with coefficients in a n-cat-over-B.
+The proof below works essentially unchanged in this case to show that the colimit is the blob complex of the total space $E$.
+}
+
+\nn{The second approach: Choose a decomposition $B = \cup X_i$
+such that the restriction of $E$ to $X_i$ is a product $F\times X_i$.
+Choose the product structure as well.
+To each codim-1 face $D_i\cap D_j$ we have a bimodule ($S^0$-module).
+And more generally to each codim-$j$ face we have an $S^{j-1}$-module.
+Decorate the decomposition with these modules and do the colimit.
+}
+
+\nn{There is a version of this last construction for arbitrary maps $E \to B$
+(not necessarily a fibration).}
+
+
+
 \subsection{A gluing theorem}
 \label{sec:gluing}
 

--- a/text/ncat.tex	Mon May 31 13:27:24 2010 -0700
+++ b/text/ncat.tex	Mon May 31 17:27:17 2010 -0700
@@ -3,7 +3,7 @@
 \def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip}
 \def\mmpar#1#2#3{\smallskip\noindent{\bf #1} (#2). {\it #3} \smallskip}
 
-\section{Definitions of $n$-categories}
+\section{$n$-categories and their modules}
 \label{sec:ncats}
 
 \subsection{Definition of $n$-categories}

--- a/text/smallblobs.tex	Mon May 31 13:27:24 2010 -0700
+++ b/text/smallblobs.tex	Mon May 31 17:27:17 2010 -0700
@@ -1,7 +1,12 @@
 %!TEX root = ../blob1.tex
 \nn{Not sure where this goes yet: small blobs, unfinished:}
 
-Fix $\cU$, an open cover of $M$. Define the `small blob complex' $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ of all blob diagrams in which every blob is contained in some open set of $\cU$. Say that an open cover $\cV$ is strictly subordinate to $\cU$ if the closure of every open set of $\cV$ is contained in some open set of $\cU$.
+Fix $\cU$, an open cover of $M$. Define the `small blob complex' $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ of all blob diagrams in which every blob is contained in some open set of $\cU$. 
+\nn{KW: We need something a little stronger: Every blob diagram (even a 0-blob diagram) is splittable into pieces which are small w.r.t.\ $\cU$.
+If field have potentially large coupons/boxes, then this is a non-trivial constraint.
+On the other hand, we could probably get away with ignoring this point.
+Maybe the exposition will be better if we sweep this technical detail under the rug?}
+Say that an open cover $\cV$ is strictly subordinate to $\cU$ if the closure of every open set of $\cV$ is contained in some open set of $\cU$.
 
 \begin{lem}
 \label{lem:CH-small-blobs}
@@ -18,14 +23,19 @@
 \todo{I think I need to understand better that proof before I can write this!}
 \end{proof}
 
-\begin{thm}[Small blobs]
+\begin{thm}[Small blobs] \label{thm:small-blobs}
 The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence.
 \end{thm}
 \begin{proof}
 We begin by describing the homotopy inverse in small degrees, to illustrate the general technique.
 We will construct a chain map $s:  \bc_*(M) \to \bc^{\cU}_*(M)$ and a homotopy $h:\bc_*(M) \to \bc_{*+1}(M)$ so that $\bdy h+h \bdy=i\circ s - \id$. The composition $s \circ i$ will just be the identity.
 
-On $0$-blobs, $s$ is just the identity; a blob diagram without any blobs is compatible with any open cover. Nevertheless, we'll begin introducing nomenclature at this point: for configuration $\beta$ of disjoint embedded balls in $M$ we'll associate a one parameter family of homeomorphisms $\phi_\beta : \Delta^1 \to \Homeo(M)$ (here $\Delta^m$ is the standard simplex $\setc{\mathbf{x} \in \Real^{m+1}}{\sum_{i=0}^m x_i = 1}$). For $0$-blobs, where $\beta = \eset$, all these homeomorphisms are just the identity.
+On $0$-blobs, $s$ is just the identity; a blob diagram without any blobs is compatible with any open cover. 
+\nn{KW: For some systems of fields this is not true.
+For example, consider a planar algebra with boxes of size greater than zero.
+So I think we should do the homotopy even in degree zero.
+But as noted above, maybe it's best to ignore this.}
+Nevertheless, we'll begin introducing nomenclature at this point: for configuration $\beta$ of disjoint embedded balls in $M$ we'll associate a one parameter family of homeomorphisms $\phi_\beta : \Delta^1 \to \Homeo(M)$ (here $\Delta^m$ is the standard simplex $\setc{\mathbf{x} \in \Real^{m+1}}{\sum_{i=0}^m x_i = 1}$). For $0$-blobs, where $\beta = \eset$, all these homeomorphisms are just the identity.
 
 When $\beta$ is a collection of disjoint embedded balls in $M$, we say that a homeomorphism of $M$ `makes $\beta$ small' if the image of each ball in $\beta$ under the homeomorphism is contained in some open set of $\cU$. Further, we'll say a homeomorphism `makes $\beta$ $\epsilon$-small' if the image of each ball is contained in some open ball of radius $\epsilon$.