--- a/text/blobdef.tex	Mon May 31 17:27:17 2010 -0700
+++ b/text/blobdef.tex	Mon May 31 23:42:37 2010 -0700
@@ -39,7 +39,7 @@
 \end{itemize}
 (See Figure \ref{blob1diagram}.)
 \begin{figure}[t]\begin{equation*}
-\mathfig{.9}{definition/single-blob}
+\mathfig{.6}{definition/single-blob}
 \end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure}
 In order to get the linear structure correct, we (officially) define
 \[
@@ -75,7 +75,7 @@
 \end{itemize}
 (See Figure \ref{blob2ddiagram}.)
 \begin{figure}[t]\begin{equation*}
-\mathfig{.9}{definition/disjoint-blobs}
+\mathfig{.6}{definition/disjoint-blobs}
 \end{equation*}\caption{A disjoint 2-blob diagram.}\label{blob2ddiagram}\end{figure}
 We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$;
 reversing the order of the blobs changes the sign.
@@ -95,7 +95,7 @@
 \end{itemize}
 (See Figure \ref{blob2ndiagram}.)
 \begin{figure}[t]\begin{equation*}
-\mathfig{.9}{definition/nested-blobs}
+\mathfig{.6}{definition/nested-blobs}
 \end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure}
 Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$
 (for some $c_1 \in \cC(B_1)$) and
@@ -153,7 +153,7 @@
 \end{itemize}
 (See Figure \ref{blobkdiagram}.)
 \begin{figure}[t]\begin{equation*}
-\mathfig{.9}{definition/k-blobs}
+\mathfig{.7}{definition/k-blobs}
 \end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure}
 
 If two blob diagrams $D_1$ and $D_2$ 

--- a/text/intro.tex	Mon May 31 17:27:17 2010 -0700
+++ b/text/intro.tex	Mon May 31 23:42:37 2010 -0700
@@ -26,15 +26,18 @@
 
 Nevertheless, when we attempt to establish all of the observed properties of the blob complex, we find this situation unsatisfactory. Thus, in the second part of the paper (section \S \ref{sec:ncats}) we pause and give yet another definition of an $n$-category, or rather a definition of an $n$-category with strong duality. (It appears that removing the duality conditions from our definition would make it more complicated rather than less.) We call these ``topological $n$-categories'', to differentiate them from previous versions. Moreover, we find that we need analogous $A_\infty$ $n$-categories, and we define these as well following very similar axioms.
 
-The basic idea is that each potential definition of an $n$-category makes a choice about the `shape' of morphisms. We try to be as lax as possible: a topological $n$-category associates a vector space to every $B$ homeomorphic to the $n$-ball. These vector spaces glue together associatively, and we require that there is an action of the homeomorphism group.
+The basic idea is that each potential definition of an $n$-category makes a choice about the `shape' of morphisms. We try to be as lax as possible: a topological $n$-category associates a vector space to every $B$ homeomorphic to the $n$-ball. These vector spaces glue together associatively, and we require that there is an action of the homeomorphism groupoid.
 For an $A_\infty$ $n$-category, we associate a chain complex instead of a vector space to each such $B$ and ask that the action of homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms. The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls labelled by a topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$.
 
-In  \S \ref{sec:ainfblob} we explain how to construct a system of fields from a topological $n$-category, and give an alternative definition of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold. Using these definitions, we show how to use the blob complex to `resolve' any topological $n$-category as an $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), in particular the `gluing formula' of Property \ref{property:gluing} below.
+In  \S \ref{sec:ainfblob} we explain how to construct a system of fields from a topological $n$-category (using a colimit along cellulations of a manifold), and give an alternative definition of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit). Using these definitions, we show how to use the blob complex to `resolve' any topological $n$-category as an $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), in particular the `gluing formula' of Property \ref{property:gluing} below.
+
+The relationship between all these ideas is sketched in Figure \ref{fig:outline}.
 
 \nn{KW: the previous two paragraphs seem a little awkward to me, but I don't presently have a good idea for fixing them.}
 
 \tikzstyle{box} = [rectangle, rounded corners, draw,outer sep = 5pt, inner sep = 5pt, line width=0.5pt]
 
+\begin{figure}[!ht]
 {\center
 
 \begin{tikzpicture}[align=center,line width = 1.5pt]
@@ -69,6 +72,9 @@
 \end{tikzpicture}
 
 }
+\caption{The main gadgets and constructions of the paper.}
+\label{fig:outline}
+\end{figure}
 
 Finally, later sections address other topics. Section \S \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, thought of as a topological $n$-category, in terms of the topology of $M$. Section \S \ref{sec:deligne} states (and in a later edition of this paper, hopefully proves) a higher dimensional generalization of the Deligne conjecture (that the little discs operad acts on Hochschild cohomology) in terms of the blob complex. The appendixes prove technical results about $\CH{M}$, and make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras.
 
@@ -167,7 +173,7 @@
 \end{property}
 As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*^\cC(X)$; this action is extended to all of $C_*(\Homeo(X))$ in Property \ref{property:evaluation} below.
 
-The blob complex is also functorial with respect to $\cC$, although we will not address this in detail here. \todo{exact w.r.t $\cC$?}
+The blob complex is also functorial (indeed, exact) with respect to $\cC$, although we will not address this in detail here.
 
 \begin{property}[Disjoint union]
 \label{property:disjoint-union}
@@ -220,19 +226,20 @@
 \end{property}
 
 In the following $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$.
-\begin{property}[$C_*(\Homeo(-))$ action]
+\begin{property}[$C_*(\Homeo(-))$ action]\mbox{}\\
+\vspace{-0.5cm}
 \label{property:evaluation}%
-There is a chain map
+\begin{enumerate}
+\item There is a chain map
 \begin{equation*}
 \ev_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X).
 \end{equation*}
 
-Restricted to $C_0(\Homeo(X))$ this is just the action of homeomorphisms described in Property \ref{property:functoriality}. 
-\nn{should probably say something about associativity here (or not?)}
+\item Restricted to $C_0(\Homeo(X))$ this is the action of homeomorphisms described in Property \ref{property:functoriality}. 
 
-For
+\item For
 any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram
-(using the gluing maps described in Property \ref{property:gluing-map}) commutes.
+(using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy).
 \begin{equation*}
 \xymatrix@C+2cm{
      \CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]^<<<<<<<<<<<<{\ev_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}}    & \bc_*(X \bigcup_Y \selfarrow) \\
@@ -241,15 +248,23 @@
             \bc_*(X) \ar[u]_{\gl_Y}
 }
 \end{equation*}
-
-\nn{unique up to homotopy?}
+\item Any such chain map satisfying points 2. and 3. above is unique, up to an iterated homotopy. (That is, any pair of homotopies have a homotopy between them, and so on.)
+\item This map is associative, in the sense that the following diagram commutes (up to homotopy).
+\begin{equation*}
+\xymatrix{
+\CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor \ev_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{\ev_X} \\
+\CH{X} \tensor \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X)
+}
+\end{equation*}
+\end{enumerate}
 \end{property}
 
-Since the blob complex is functorial in the manifold $X$, we can use this to build a chain map
+Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps
 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$
+for any homeomorphic pair $X$ and $Y$, 
 satisfying corresponding conditions.
 
-In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields. Below, we talk about the blob complex associated to a topological $n$-category, implicitly passing to the system of fields. Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category.
+In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields. Below, we talk about the blob complex associated to a topological $n$-category, implicitly passing first to the system of fields. Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category.
 
 \begin{property}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category]
 \label{property:blobs-ainfty}

--- a/text/ncat.tex	Mon May 31 17:27:17 2010 -0700
+++ b/text/ncat.tex	Mon May 31 23:42:37 2010 -0700
@@ -95,7 +95,7 @@
 Morphisms are modeled on balls, so their boundaries are modeled on spheres.
 In other words, we need to extend the functors $\cC_{k-1}$ from balls to spheres, for 
 $1\le k \le n$.
-At first might seem that we need another axiom for this, but in fact once we have
+At first it might seem that we need another axiom for this, but in fact once we have
 all the axioms in the subsection for $0$ through $k-1$ we can use a coend
 construction, as described in Subsection \ref{ss:ncat-coend} below, to extend $\cC_{k-1}$
 to spheres (and any other manifolds):
@@ -107,6 +107,7 @@
 homeomorphisms to the category of sets and bijections.
 \end{prop}
 
+We postpone the proof of this result until after we've actually given all the axioms. Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, along with the data described in other Axioms at lower levels. 
 
 %In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point.
 
@@ -515,6 +516,8 @@
 (Note that homotopy invariance implies isotopy invariance.)
 For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to
 be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection.
+
+Recall we described a system of fields and local relations based on maps to $T$ in Example \ref{ex:maps-to-a-space(fields)} above. Constructing a system of fields from $\pi_{\leq n}(T)$ recovers that example.
 \end{example}
 
 \begin{example}[Maps to a space, with a fiber]
@@ -556,6 +559,9 @@
 Define $\cC(X; c)$, for $X$ an $n$-ball,
 to be the dual Hilbert space $A(X\times F; c)$.
 \nn{refer elsewhere for details?}
+
+
+Recall we described a system of fields and local relations based on a `traditional $n$-category' $C$ in Example \ref{ex:traditional-n-categories(fields)} above. Constructing a system of fields from $\cC$ recovers that example.
 \end{example}
 
 Finally, we describe a version of the bordism $n$-category suitable to our definitions.

--- a/text/tqftreview.tex	Mon May 31 17:27:17 2010 -0700
+++ b/text/tqftreview.tex	Mon May 31 23:42:37 2010 -0700
@@ -30,14 +30,20 @@
 Before finishing the definition of fields, we give two motivating examples
 (actually, families of examples) of systems of fields.
 
-The first examples: Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps
+\begin{example}
+\label{ex:maps-to-a-space(fields)}
+Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps
 from X to $B$.
+\end{example}
 
-The second examples: Fix an $n$-category $C$, and let $\cC(X)$ be 
+\begin{example}
+\label{ex:traditional-n-categories(fields)}
+Fix an $n$-category $C$, and let $\cC(X)$ be 
 the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by
 $j$-morphisms of $C$.
 One can think of such sub-cell-complexes as dual to pasting diagrams for $C$.
 This is described in more detail below.
+\end{example}
 
 Now for the rest of the definition of system of fields.
 \begin{enumerate}
@@ -262,8 +268,23 @@
 
 \subsection{Local relations}
 \label{sec:local-relations}
+Local relations are certain subspaces of the fields on balls, which form an ideal under gluing. Again, we give the examples first.
 
+\addtocounter{prop}{-2}
+\begin{example}[contd.]
+For maps into spaces, $U(B; c)$ is generated by fields of the form $a-b \in \lf(B; c)$,
+where $a$ and $b$ are maps (fields) which are homotopic rel boundary.
+\end{example}
 
+\begin{example}[contd.]
+For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map
+$\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into
+domain and range.
+\end{example}
+
+These motivate the following definition.
+
+\begin{defn}
 A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$,
 for all $n$-manifolds $B$ which are
 homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$, 
@@ -277,17 +298,9 @@
 \item ideal with respect to gluing:
 if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\bullet r \in U(B)$
 \end{enumerate}
-See \cite{kw:tqft} for details.
-
-
-For maps into spaces, $U(B; c)$ is generated by things of the form $a-b \in \lf(B; c)$,
-where $a$ and $b$ are maps (fields) which are homotopic rel boundary.
+\end{defn}
+See \cite{kw:tqft} for further details.
 
-For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map
-$\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into
-domain and range.
-
-\nn{maybe examples of local relations before general def?}
 
 \subsection{Constructing a TQFT}
 \label{sec:constructing-a-tqft}