# HG changeset patch
# User Scott Morrison <scott@tqft.net>
# Date 1277595088 25200
# Node ID a02a6158f3bddc4db1131c797bd851d9356a23e4
# Parent  979fbe9a14e849689a86b215da7516596e6f37b3
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.

diff -r 979fbe9a14e8 -r a02a6158f3bd text/a_inf_blob.tex
--- 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{
diff -r 979fbe9a14e8 -r a02a6158f3bd text/blobdef.tex
--- 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 
diff -r 979fbe9a14e8 -r a02a6158f3bd text/comm_alg.tex
--- 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}
diff -r 979fbe9a14e8 -r a02a6158f3bd text/deligne.tex
--- 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
diff -r 979fbe9a14e8 -r a02a6158f3bd text/evmap.tex
--- 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{
diff -r 979fbe9a14e8 -r a02a6158f3bd text/hochschild.tex
--- 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}
diff -r 979fbe9a14e8 -r a02a6158f3bd text/intro.tex
--- 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.
 
diff -r 979fbe9a14e8 -r a02a6158f3bd text/ncat.tex
--- 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'