--- a/text/a_inf_blob.tex	Fri Jun 25 09:48:24 2010 -0700
+++ b/text/a_inf_blob.tex	Sat Jun 26 16:31:28 2010 -0700
@@ -42,7 +42,7 @@
 
 \nn{need to settle on notation; proof and statement are inconsistent}
 
-\begin{thm} \label{product_thm}
+\begin{thm} \label{thm:product}
 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*}
@@ -57,7 +57,7 @@
 \end{thm}
 
 
-\begin{proof}%[Proof of Theorem \ref{product_thm}]
+\begin{proof}
 We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}.
 
 First we define a map 
@@ -214,7 +214,7 @@
 collection of acyclic subcomplexes, so by the usual MoAM argument these two maps
 are homotopic.
 
-This concludes the proof of Theorem \ref{product_thm}.
+This concludes the proof of Theorem \ref{thm:product}.
 \end{proof}
 
 \nn{need to prove a version where $E$ above has dimension $m<n$; result is an $n{-}m$-category}
@@ -227,12 +227,12 @@
 The new-fangled and old-fashioned blob complexes are homotopic.
 \end{cor}
 \begin{proof}
-Apply Theorem \ref{product_thm} with the fiber $F$ equal to a point.
+Apply Theorem \ref{thm:product} with the fiber $F$ equal to a point.
 \end{proof}
 
 \medskip
 
-Theorem \ref{product_thm} extends to the case of general fiber bundles
+Theorem \ref{thm:product} extends to the case of general fiber bundles
 \[
 	F \to E \to Y .
 \]
@@ -247,7 +247,7 @@
 Let $\cF_E$ denote this $k$-category over $Y$.
 We can adapt the homotopy colimit construction (based decompositions of $Y$ into balls) to
 get a chain complex $\cF_E(Y)$.
-The proof of Theorem \ref{product_thm} goes through essentially unchanged 
+The proof of Theorem \ref{thm:product} goes through essentially unchanged 
 to show that
 \[
 	\bc_*(E) \simeq \cF_E(Y) .
@@ -298,9 +298,9 @@
 
 \begin{proof}
 \nn{for now, just prove $k=0$ case.}
-The proof is similar to that of Theorem \ref{product_thm}.
+The proof is similar to that of Theorem \ref{thm:product}.
 We give a short sketch with emphasis on the differences from 
-the proof of Theorem \ref{product_thm}.
+the proof of Theorem \ref{thm:product}.
 
 Let $\cT$ denote the chain complex $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.
 Recall that this is a homotopy colimit based on decompositions of the interval $J$.
@@ -316,17 +316,15 @@
 a subcomplex of $G_*$. 
 
 Next we define a map $\phi:G_*\to \cT$ using the method of acyclic models.
-As in the proof of Theorem \ref{product_thm}, we assign to a generator $a$ of $G_*$
+As in the proof of Theorem \ref{thm:product}, we assign to a generator $a$ of $G_*$
 an acyclic subcomplex which is (roughly) $\psi\inv(a)$.
 The proof of acyclicity is easier in this case since any pair of decompositions of $J$ have
 a common refinement.
 
 The proof that these two maps are inverse to each other is the same as in
-Theorem \ref{product_thm}.
+Theorem \ref{thm:product}.
 \end{proof}
 
-This establishes Property \ref{property:gluing}.
-
 \noop{
 Let $\cT$ denote the $n{-}k$-category $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.
 Let $D$ be an $n{-}k$-ball.
@@ -337,13 +335,14 @@
 on $D\times X$, a decomposition of $J$ such that $b$ splits on the corresponding
 decomposition of $D\times X$.
 The proof that these two maps are inverse to each other is the same as in
-Theorem \ref{product_thm}.
+Theorem \ref{thm:product}.
 }
 
 
 \medskip
 
 \subsection{Reconstructing mapping spaces}
+\label{sec:map-recon}
 
 The next theorem shows how to reconstruct a mapping space from local data.
 Let $T$ be a topological space, let $M$ be an $n$-manifold, 
@@ -353,7 +352,8 @@
 want to know about spaces of maps of $k$-balls into $T$ ($k\le n$).
 To simplify notation, let $\cT = \pi^\infty_{\leq n}(T)$.
 
-\begin{thm} \label{thm:map-recon}
+\begin{thm}
+\label{thm:map-recon}
 The blob complex for $M$ with coefficients in the fundamental $A_\infty$ $n$-category for $T$ 
 is quasi-isomorphic to singular chains on maps from $M$ to $T$.
 $$\cB^\cT(M) \simeq C_*(\Maps(M\to T)).$$
@@ -369,7 +369,7 @@
 \end{rem}
 
 \begin{proof}
-The proof is again similar to that of Theorem \ref{product_thm}.
+The proof is again similar to that of Theorem \ref{thm:product}.
 
 We begin by constructing chain map $\psi: \cB^\cT(M) \to C_*(\Maps(M\to T))$.
 
@@ -411,7 +411,7 @@
 Define $D(a)$ to be the subcomplex of $\cB^\cT(M)$ generated by all 
 pairs $(b, \ol{K})$, where $b$ is a generator appearing in an iterated boundary of $a$
 and $\ol{K}$ is an index of the homotopy colimit $\cB^\cT(M)$.
-(See the proof of Theorem \ref{product_thm} for more details.)
+(See the proof of Theorem \ref{thm:product} for more details.)
 The same proof as of Lemma \ref{lem:d-a-acyclic} shows that $D(a)$ is acyclic.
 By the usual acyclic models nonsense, there is a (unique up to homotopy)
 map $\phi:G_*\to \cB^\cT(M)$ such that $\phi(a)\in D(a)$.
@@ -423,7 +423,7 @@
 
 It is now easy to see that $\psi\circ\phi$ is the identity on the nose.
 Another acyclic models argument shows that $\phi\circ\psi$ is homotopic to the identity.
-(See the proof of Theorem \ref{product_thm} for more details.)
+(See the proof of Theorem \ref{thm:product} for more details.)
 \end{proof}
 
 \noop{

--- a/text/blobdef.tex	Fri Jun 25 09:48:24 2010 -0700
+++ b/text/blobdef.tex	Sat Jun 26 16:31:28 2010 -0700
@@ -58,7 +58,7 @@
 
 Note that the skein space $A(X)$
 is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$.
-This is Property \ref{property:skein-modules}, and also used in the second 
+This is Theorem \ref{thm:skein-modules}, and also used in the second 
 half of Property \ref{property:contractibility}.
 
 Next, we want the vector space $\bc_2(X)$ to capture `the space of all relations 

--- a/text/comm_alg.tex	Fri Jun 25 09:48:24 2010 -0700
+++ b/text/comm_alg.tex	Sat Jun 26 16:31:28 2010 -0700
@@ -105,7 +105,7 @@
 
 \medskip
 
-In view of \ref{hochthm}, we have proved that $HH_*(k[t]) \cong C_*(\Sigma^\infty(S^1), k)$,
+In view of Theorem \ref{thm:hochschild}, we have proved that $HH_*(k[t]) \cong C_*(\Sigma^\infty(S^1), k)$,
 and that the cyclic homology of $k[t]$ is related to the action of rotations
 on $C_*(\Sigma^\infty(S^1), k)$.
 \nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section}

--- a/text/deligne.tex	Fri Jun 25 09:48:24 2010 -0700
+++ b/text/deligne.tex	Sat Jun 26 16:31:28 2010 -0700
@@ -206,8 +206,8 @@
 The main result of this section is that this chain map extends to the full singular
 chain complex $C_*(FG^n_{\ol{M}\ol{N}})$.
 
-\begin{prop}
-\label{prop:deligne}
+\begin{thm}
+\label{thm:deligne}
 There is a collection of chain maps
 \[
 	C_*(FG^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes 
@@ -216,7 +216,7 @@
 which satisfy the operad compatibility conditions.
 On $C_0(FG^n_{\ol{M}\ol{N}})$ this agrees with the chain map $p$ defined above.
 When $k=0$, this coincides with the $C_*(\Homeo(M_0\to N_0))$ action of Section \ref{sec:evaluation}.
-\end{prop}
+\end{thm}
 
 If, in analogy to Hochschild cochains, we define elements of $\hom(M, N)$
 to be ``blob cochains", we can summarize the above proposition by saying that the $n$-FG operad acts on

--- a/text/evmap.tex	Fri Jun 25 09:48:24 2010 -0700
+++ b/text/evmap.tex	Sat Jun 26 16:31:28 2010 -0700
@@ -21,7 +21,7 @@
 such that
 \begin{enumerate}
 \item on $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of 
-$\Homeo(X, Y)$ on $\bc_*(X)$ (Proposition (\ref{diff0prop})), and
+$\Homeo(X, Y)$ on $\bc_*(X)$ (Property (\ref{property:functoriality})), and
 \item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, 
 the following diagram commutes up to homotopy
 \eq{ \xymatrix{

--- a/text/hochschild.tex	Fri Jun 25 09:48:24 2010 -0700
+++ b/text/hochschild.tex	Sat Jun 26 16:31:28 2010 -0700
@@ -66,7 +66,8 @@
 so it suffices to show that they are quasi-isomorphic.
 
 We claim that
-\begin{thm} \label{hochthm}
+\begin{thm}
+\label{thm:hochschild}
 The blob complex $\bc_*(S^1; C)$ on the circle is homotopy equivalent to the
 usual Hochschild complex for $C$.
 \end{thm}

--- a/text/intro.tex	Fri Jun 25 09:48:24 2010 -0700
+++ b/text/intro.tex	Sat Jun 26 16:31:28 2010 -0700
@@ -3,13 +3,13 @@
 \section{Introduction}
 
 We construct the ``blob complex'' $\bc_*(M; \cC)$ associated to an $n$-manifold $M$ and a ``linear $n$-category with strong duality'' $\cC$.
-This blob complex provides a simultaneous generalisation of several well-understood constructions:
+This blob complex provides a simultaneous generalization of several well-understood constructions:
 \begin{itemize}
 \item The vector space $H_0(\bc_*(M; \cC))$ is isomorphic to the usual topological quantum field theory invariant of $M$ associated to $\cC$.
-(See Property \ref{property:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.)
+(See Theorem \ref{thm:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.)
 \item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra), 
 the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$.
-(See Property \ref{property:hochschild} and \S \ref{sec:hochschild}.)
+(See Theorem \ref{thm:hochschild} and \S \ref{sec:hochschild}.)
 \item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category (see \S \ref{sec:comm_alg}), we have 
 that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains
 on the configuration space of unlabeled points in $M$.
@@ -23,16 +23,16 @@
 
 The blob complex has good formal properties, summarized in \S \ref{sec:properties}.
 These include an action of $\CH{M}$, 
-extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (see Property \ref{property:evaluation}) and a gluing 
-formula allowing calculations by cutting manifolds into smaller parts (see Property \ref{property:gluing}).
+extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (see Theorem \ref{thm:evaluation}) and a gluing 
+formula allowing calculations by cutting manifolds into smaller parts (see Theorem \ref{thm:gluing}).
 
 We expect applications of the blob complex to contact topology and Khovanov homology but do not address these in this paper.
 See \S \ref{sec:future} for slightly more detail.
 
 \subsubsection{Structure of the paper}
-The three subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}), 
-summarise the formal properties of the blob complex (see \S \ref{sec:properties}) 
-and outline anticipated future directions and applications (see \S \ref{sec:future}).
+The subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}), 
+summarise the formal properties of the blob complex (see \S \ref{sec:properties}), describe known specializations (see \S \ref{sec:specializations}), outline the major results of the paper (see \S \ref{sec:structure} and \S \ref{sec:applications})
+and outline anticipated future directions (see \S \ref{sec:future}).
 
 The first part of the paper (sections \S \ref{sec:fields}---\S \ref{sec:evaluation}) gives the definition of the blob complex, 
 and establishes some of its properties.
@@ -64,7 +64,7 @@
 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 particular the `gluing formula' of Theorem \ref{thm:gluing} below.
 
 The relationship between all these ideas is sketched in Figure \ref{fig:outline}.
 
@@ -101,7 +101,7 @@
 	Example \ref{ex:traditional-n-categories(fields)} \\ and \S \ref{ss:ncat_fields}
 	%$\displaystyle \cF(M) = \DirectSum_{c \in\cell(M)} \cC(c)$ \\ $\displaystyle \cU(B) = \DirectSum_{c \in \cell(B)} \ker \ev: \cC(c) \to \cC(B)$
    } (FU);
-\draw[->] (BC) -- node[left] {$H_0$} node[right] {c.f. Property \ref{property:skein-modules}} (A);
+\draw[->] (BC) -- node[left] {$H_0$} node[right] {c.f. Theorem \ref{thm:skein-modules}} (A);
 
 \draw[->] (FU) -- node[left] {blob complex \\ for balls} (Cs);
 \draw (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs);
@@ -217,7 +217,7 @@
 complexes and isomorphisms between them.
 \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.
+this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:evaluation} below.
 
 The blob complex is also functorial (indeed, exact) with respect to $\cC$, 
 although we will not address this in detail here.
@@ -256,8 +256,17 @@
 \end{equation}
 \end{property}
 
-\begin{property}[Skein modules]
-\label{property:skein-modules}%
+Properties \ref{property:functoriality} will be immediate from the definition given in
+\S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there.
+Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}.
+
+\subsection{Specializations}
+\label{sec:specializations}
+
+The blob complex is a simultaneous generalization of the TQFT skein module construction and of Hochschild homology.
+
+\begin{thm}[Skein modules]
+\label{thm: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 $\cC$.
@@ -265,23 +274,30 @@
 \begin{equation*}
 H_0(\bc_*^{\cC}(X)) \iso A^{\cC}(X)
 \end{equation*}
-\end{property}
+\end{thm}
 
-\todo{Somehow, the Hochschild homology thing isn't a "property".
-Let's move it and call it a theorem? -S}
-\begin{property}[Hochschild homology when $X=S^1$]
-\label{property:hochschild}%
+\newtheorem*{thm:hochschild}{Theorem \ref{thm:hochschild}}
+
+\begin{thm:hochschild}[Hochschild homology when $X=S^1$]
 The blob complex for a $1$-category $\cC$ on the circle is
 quasi-isomorphic to the Hochschild complex.
 \begin{equation*}
 \xymatrix{\bc_*^{\cC}(S^1) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & \HC_*(\cC).}
 \end{equation*}
-\end{property}
+\end{thm:hochschild}
+
+Theorem \ref{thm:skein-modules} is immediate from the definition, and
+Theorem \ref{thm:hochschild} is established in \S \ref{sec:hochschild}.
+We also note Appendix \ref{sec:comm_alg} which describes the blob complex when $\cC$ is a one of certain commutative algebras thought of as an $n$-category.
+
+
+\subsection{Structure of the blob complex}
+\label{sec:structure}
 
 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]\mbox{}\\
+\begin{thm}[$C_*(\Homeo(-))$ action]\mbox{}\\
 \vspace{-0.5cm}
-\label{property:evaluation}%
+\label{thm:evaluation}%
 \begin{enumerate}
 \item There is a chain map
 \begin{equation*}
@@ -311,7 +327,7 @@
 }
 \end{equation*}
 \end{enumerate}
-\end{property}
+\end{thm}
 
 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps
 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$
@@ -322,8 +338,8 @@
 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}
+\begin{thm}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category]
+\label{thm:blobs-ainfty}
 Let $\cC$ be  a topological $n$-category.
 Let $Y$ be an $n{-}k$-manifold. 
 There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, 
@@ -331,8 +347,8 @@
 $$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ 
 (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) 
 These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in 
-Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Property \ref{property:evaluation}.
-\end{property}
+Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}.
+\end{thm}
 \begin{rem}
 Perhaps the most interesting case is when $Y$ is just a point; then we have a way of building an $A_\infty$ $n$-category from a topological $n$-category.
 We think of this $A_\infty$ $n$-category as a free resolution.
@@ -342,24 +358,26 @@
 instead of a topological $n$-category; this is described in \S \ref{sec:ainfblob}.
 The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$.
 
-\begin{property}[Product formula]
-\label{property:product}
+\newtheorem*{thm:product}{Theorem \ref{thm:product}}
+
+\begin{thm:product}[Product formula]
 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold.
 Let $\cC$ be an $n$-category.
-Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Property \ref{property:blobs-ainfty}).
+Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Theorem \ref{thm:blobs-ainfty}).
 Then
 \[
 	\bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W).
 \]
-\end{property}
+\end{thm:product}
 We also give a generalization of this statement for arbitrary fibre bundles, in \S \ref{moddecss}, and a sketch of a statement for arbitrary maps.
 
 Fix a topological $n$-category $\cC$, which we'll omit from the notation.
 Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category.
 (See Appendix \ref{sec:comparing-A-infty} for the translation between topological $A_\infty$ $1$-categories and the usual algebraic notion of an $A_\infty$ category.)
 
-\begin{property}[Gluing formula]
-\label{property:gluing}%
+\newtheorem*{thm:gluing}{Theorem \ref{thm:gluing}}
+
+\begin{thm:gluing}[Gluing formula]
 \mbox{}% <-- gets the indenting right
 \begin{itemize}
 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an
@@ -371,32 +389,37 @@
 \bc_*(X_\text{glued}) \simeq \bc_*(X_\text{cut}) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow
 \end{equation*}
 \end{itemize}
-\end{property}
+\end{thm:gluing}
+
+Theorem \ref{thm:evaluation} is proved in
+in \S \ref{sec:evaluation}, Theorem \ref{thm:blobs-ainfty} appears as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats},
+and Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, with Theorem \ref{thm:gluing} then a relatively straightforward consequence of the proof, explained in \S \ref{sec:gluing}.
 
-Finally, we prove two theorems which we consider as applications.
+\subsection{Applications}
+\label{sec:applications}
+Finally, we give two theorems which we consider as applications.
 
-\begin{thm}[Mapping spaces]
+\newtheorem*{thm:map-recon}{Theorem \ref{thm:map-recon}}
+
+\begin{thm:map-recon}[Mapping spaces]
 Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps 
 $B^n \to T$.
 (The case $n=1$ is the usual $A_\infty$-category of paths in $T$.)
 Then 
 $$\bc_*(X, \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$
-\end{thm}
+\end{thm:map-recon}
 
-This says that we can recover the (homotopic) space of maps to $T$ via blob homology from local data.
+This says that we can recover the (homotopic) space of maps to $T$ via blob homology from local data. The proof appears in \S \ref{sec:map-recon}.
 
-\begin{thm}[Higher dimensional Deligne conjecture]
-\label{thm:deligne}
+\newtheorem*{thm:deligne}{Theorem \ref{thm:deligne}}
+
+\begin{thm:deligne}[Higher dimensional Deligne conjecture]
 The singular chains of the $n$-dimensional fat graph operad act on blob cochains.
-\end{thm}
+\end{thm:deligne}
 See \S \ref{sec:deligne} for a full explanation of the statement, and an outline of the proof.
 
-Properties \ref{property:functoriality} and \ref{property:skein-modules} will be immediate from the definition given in
-\S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there.
-Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}.
-Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} 
-in \S \ref{sec:evaluation}, Property \ref{property:blobs-ainfty} as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats},
-and Properties \ref{property:product} and \ref{property:gluing} in \S \ref{sec:ainfblob} as consequences of Theorem \ref{product_thm}.
+
+
 
 \subsection{Future directions}
 \label{sec:future}
@@ -425,6 +448,6 @@
 
 \subsection{Thanks and acknowledgements}
 We'd like to thank David Ben-Zvi, Kevin Costello, Chris Douglas,
-Michael Freedman, Vaughan Jones, Justin Roberts, Chris Schommer-Pries, Peter Teichner \nn{and who else?} for many interesting and useful conversations. 
+Michael Freedman, Vaughan Jones, Alexander Kirillov, Justin Roberts, Chris Schommer-Pries, Peter Teichner \nn{and who else?} for many interesting and useful conversations. 
 During this work, Kevin Walker has been at Microsoft Station Q, and Scott Morrison has been at Microsoft Station Q and the Miller Institute for Basic Research at UC Berkeley.
 

--- a/text/ncat.tex	Fri Jun 25 09:48:24 2010 -0700
+++ b/text/ncat.tex	Sat Jun 26 16:31:28 2010 -0700
@@ -789,7 +789,7 @@
 where $\bc^\cE_*$ denotes the blob complex based on $\cE$.
 \end{example}
 
-This example will be essential for Theorem \ref{product_thm} below, which allows us to compute the blob complex of a product.
+This example will be essential for Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product.
 Notice that with $F$ a point, the above example is a construction turning a topological 
 $n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$.
 We think of this as providing a `free resolution'