Binary file diagrams/definition/single-blob.pdf has changed

--- a/text/basic_properties.tex	Mon Jul 12 17:29:25 2010 -0600
+++ b/text/basic_properties.tex	Wed Jul 14 11:06:11 2010 -0600
@@ -6,8 +6,8 @@
 In this section we complete the proofs of Properties 2-4.
 Throughout the paper, where possible, we prove results using Properties 1-4, 
 rather than the actual definition of blob homology.
-This allows the possibility of future improvements to or alternatives on our definition.
-In fact, we hope that there may be a characterisation of blob homology in 
+This allows the possibility of future improvements on or alternatives to our definition.
+In fact, we hope that there may be a characterization of the blob complex in 
 terms of Properties 1-4, but at this point we are unaware of one.
 
 Recall Property \ref{property:disjoint-union}, 
@@ -67,10 +67,8 @@
 This follows from Properties \ref{property:disjoint-union} and \ref{property:contractibility}.
 \end{proof}
 
-Define the {\it support} of a blob diagram to be the union of all the 
+Recall the definition of the support of a blob diagram as the union of all the 
 blobs of the diagram.
-Define the support of a linear combination of blob diagrams to be the union of the 
-supports of the constituent diagrams.
 For future use we prove the following lemma.
 
 \begin{lemma} \label{support-shrink}
@@ -93,9 +91,7 @@
 \end{proof}
 
 For the next proposition we will temporarily restore $n$-manifold boundary
-conditions to the notation.
-
-Let $X$ be an $n$-manifold, $\bd X = Y \cup Y \cup Z$.
+conditions to the notation. Let $X$ be an $n$-manifold, with $\bd X = Y \cup Y \cup Z$.
 Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$
 with boundary $Z\sgl$.
 Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $Y$ and $Z$,
@@ -103,6 +99,7 @@
 If $b = a$, then we can glue up blob diagrams on
 $X$ to get blob diagrams on $X\sgl$.
 This proves Property \ref{property:gluing-map}, which we restate here in more detail.
+\todo{This needs more detail, because this is false without careful attention to non-manifold components, etc.}
 
 \textbf{Property \ref{property:gluing-map}.}\emph{
 There is a natural chain map

--- a/text/blobdef.tex	Mon Jul 12 17:29:25 2010 -0600
+++ b/text/blobdef.tex	Wed Jul 14 11:06:11 2010 -0600
@@ -4,38 +4,37 @@
 \label{sec:blob-definition}
 
 Let $X$ be an $n$-manifold.
-Let $\cC$ be a fixed system of fields (enriched over Vect) and local relations.
-(If $\cC$ is not enriched over Vect, we can make it so by allowing finite
-linear combinations of elements of $\cC(X; c)$, for fixed $c\in \cC(\bd X)$.)
+Let $\cC$ be a fixed system of fields and local relations.
+We'll assume it is enriched over \textbf{Vect}, and if it is not we can make it so by allowing finite
+linear combinations of elements of $\cC(X; c)$, for fixed $c\in \cC(\bd X)$.
 
-In this section we will usually suppress boundary conditions on $X$ from the notation
-(e.g. write $\lf(X)$ instead of $\lf(X; c)$).
+In this section we will usually suppress boundary conditions on $X$ from the notation, e.g. by writing $\lf(X)$ instead of $\lf(X; c)$.
 
 We want to replace the quotient
 \[
 	A(X) \deq \lf(X) / U(X)
 \]
-of the previous section with a resolution
+of Definition \ref{defn:TQFT-invariant} with a resolution
 \[
 	\cdots \to \bc_2(X) \to \bc_1(X) \to \bc_0(X) .
 \]
 
-We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$.
+We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$.   \todo{create a numbered definition for the general case}
 
 We of course define $\bc_0(X) = \lf(X)$.
 (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$.
 We'll omit this sort of detail in the rest of this section.)
-In other words, $\bc_0(X)$ is just the vector space of fields on $X$.
+In other words, $\bc_0(X)$ is just the vector space of all fields on $X$.
 
 We want the vector space $\bc_1(X)$ to capture `the space of all local relations that can be imposed on $\bc_0(X)$'.
-Thus we say  a $1$-blob diagram consists of
+Thus we say  a $1$-blob diagram consists of:
 \begin{itemize}
 \item An embedded closed ball (``blob") $B \sub X$.
 \item A boundary condition $c \in \cC(\bdy B) = \cC(\bd(X \setmin B))$.
 \item A field $r \in \cC(X \setmin B; c)$.
 \item A local relation field $u \in U(B; c)$.
 \end{itemize}
-(See Figure \ref{blob1diagram}.)
+(See Figure \ref{blob1diagram}.) Since $c$ is implicitly determined by $u$ or $r$, we usually omit it from the notation.
 \begin{figure}[t]\begin{equation*}
 \mathfig{.6}{definition/single-blob}
 \end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure}
@@ -56,15 +55,18 @@
 just erasing the blob from the picture
 (but keeping the blob label $u$).
 
-Note that the skein space $A(X)$
-is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$.
-This is Theorem \ref{thm:skein-modules}, and also used in the second 
+Note that directly from the definition we have
+\begin{thm}
+\label{thm:skein-modules}
+The skein module $A(X)$ is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$.
+\end{thm}
+This also establishes the second 
 half of Property \ref{property:contractibility}.
 
 Next, we want the vector space $\bc_2(X)$ to capture `the space of all relations 
 (redundancies, syzygies) among the 
 local relations encoded in $\bc_1(X)$'.
-More specifically, a $2$-blob diagram, comes in one of two types, disjoint and nested.
+A $2$-blob diagram, comes in one of two types, disjoint and nested.
 A disjoint 2-blob diagram consists of
 \begin{itemize}
 \item A pair of closed balls (blobs) $B_1, B_2 \sub X$ with disjoint interiors.
@@ -98,12 +100,11 @@
 \mathfig{.6}{definition/nested-blobs}
 \end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure}
 Define $\bd(B_1, B_2, u, r', r) = (B_2, u\bullet r', r) - (B_1, u, r' \bullet r)$.
-Note that the requirement that
-local relations are an ideal with respect to gluing guarantees that $u\bullet r' \in U(B_2)$.
 As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating
 sum of the two ways of erasing one of the blobs.
 When  we erase the inner blob, the outer blob inherits the label $u\bullet r'$.
-It is again easy to check that $\bd^2 = 0$.
+It is again easy to check that $\bd^2 = 0$. Note that the requirement that
+local relations are an ideal with respect to gluing guarantees that $u\bullet r' \in U(B_2)$.
 
 As with the $1$-blob diagrams, in order to get the linear structure correct the actual definition is 
 \begin{eqnarray*}
@@ -117,8 +118,8 @@
 			U(B_1; c_1) \otimes \lf(B_2 \setmin B_1; c_1) \tensor \cC(X \setminus B_2; c_2)
 	\right) .
 \end{eqnarray*}
-For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign
-(rather than a new, linearly independent 2-blob diagram). 
+For the disjoint blobs, reversing the ordering of $B_1$ and $B_2$ introduces a minus sign
+(rather than a new, linearly independent, 2-blob diagram). 
 \noop{
 \nn{Hmm, I think we should be doing this for nested blobs too -- 
 we shouldn't force the linear indexing of the blobs to have anything to do with 
@@ -157,7 +158,7 @@
 \item A field $r \in \cC(X \setmin B^t; c^t)$,
 where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$
 is determined by the $c_i$'s.
-$r$ is required to be splittable along the boundaries of all blobs, twigs or not.
+$r$ is required to be splittable along the boundaries of all blobs, twigs or not. (This is equivalent to asking for a field on of the components of $X \setmin B^t$.)
 \item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$,
 where $c_j$ is the restriction of $c^t$ to $\bd B_j$.
 If $B_i = B_j$ then $u_i = u_j$.
@@ -171,12 +172,12 @@
 differ only by a reordering of the blobs, then we identify
 $D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$.
 
-$\bc_k(X)$ is, roughly, all finite linear combinations of $k$-blob diagrams.
+Roughly, then, $\bc_k(X)$ is all finite linear combinations of $k$-blob diagrams.
 As before, the official definition is in terms of direct sums
 of tensor products:
 \[
 	\bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}}
-		\left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) .
+		\left( \bigotimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) .
 \]
 Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above.
 The index $\overline{c}$ runs over all boundary conditions, again as described above and $j$ runs over all indices of twig blobs.
@@ -190,9 +191,9 @@
 Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram.
 Let $E_j(b)$ denote the result of erasing the $j$-th blob.
 If $B_j$ is not a twig blob, this involves only decrementing
-the indices of blobs $B_{j+1},\ldots,B_{k-1}$.
+the indices of blobs $B_{j+1},\ldots,B_{k}$.
 If $B_j$ is a twig blob, we have to assign new local relation labels
-if removing $B_j$ creates new twig blobs.
+if removing $B_j$ creates new twig blobs. \todo{Have to say what happens when no new twig blobs are created}
 If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$,
 where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$.
 Finally, define
@@ -203,7 +204,7 @@
 Thus we have a chain complex.
 
 Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, is immediately obvious from the definition.
-A homeomorphism acts in an obvious on blobs and on fields.
+A homeomorphism acts in an obvious way on blobs and on fields.
 
 We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, 
 to be the union of the blobs of $b$.
@@ -225,8 +226,8 @@
 \end{itemize}
 For example, a diagram of $k$ strictly nested blobs corresponds to a $k$-simplex, while
 a diagram of $k$ disjoint blobs corresponds to a $k$-cube.
-(This correspondence works best if we thing of each twig label $u_i$ as having the form
+(This correspondence works best if we think of each twig label $u_i$ as having the form
 $x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cC(B_i) \to C$ is the evaluation map, 
-and $s:C \to \cC(B_i)$ is some fixed section of $e$.)
+and $s:C \to \cC(B_i)$ is some fixed section of $e$. \todo{This parenthetical remark mysteriously specialises to the category case})
 
 

--- a/text/comm_alg.tex	Mon Jul 12 17:29:25 2010 -0600
+++ b/text/comm_alg.tex	Wed Jul 14 11:06:11 2010 -0600
@@ -100,7 +100,7 @@
 
 \begin{proof}
 The actions agree in degree 0, and both are compatible with gluing.
-(cf. uniqueness statement in \ref{CHprop}.)
+(cf. uniqueness statement in Theorem \ref{thm:CH}.)
 \end{proof}
 
 \medskip

--- a/text/deligne.tex	Mon Jul 12 17:29:25 2010 -0600
+++ b/text/deligne.tex	Wed Jul 14 11:06:11 2010 -0600
@@ -227,7 +227,7 @@
 \begin{proof}
 As described above, $FG^n_{\overline{M}, \overline{N}}$ is equal to the disjoint
 union of products of homeomorphism spaces, modulo some relations.
-By Proposition \ref{CHprop} and the Eilenberg-Zilber theorem, we have for each such product $P$
+By Theorem \ref{thm:CH} and the Eilenberg-Zilber theorem, we have for each such product $P$
 a chain map
 \[
 	C_*(P)\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes 

--- a/text/evmap.tex	Mon Jul 12 17:29:25 2010 -0600
+++ b/text/evmap.tex	Wed Jul 14 11:06:11 2010 -0600
@@ -13,7 +13,7 @@
 than simplices --- they can be based on any linear polyhedron.
 \nn{be more restrictive here?  does more need to be said?})
 
-\begin{prop}  \label{CHprop}
+\begin{thm}  \label{thm:CH}
 For $n$-manifolds $X$ and $Y$ there is a chain map
 \eq{
     e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y)
@@ -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 $\bc_*(X)$ (Property (\ref{property:functoriality})), and
+$\Homeo(X, Y)$ on $\bc_*(X)$  described in 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
 \begin{equation*}
@@ -35,7 +35,7 @@
 \end{enumerate}
 Up to (iterated) homotopy, there is a unique family $\{e_{XY}\}$ of chain maps
 satisfying the above two conditions.
-\end{prop}
+\end{thm}
 
 Before giving the proof, we state the essential technical tool of Lemma \ref{extension_lemma}, 
 and then give an outline of the method of proof.
@@ -75,7 +75,7 @@
 
 \medskip
 
-Before diving into the details, we outline our strategy for the proof of Proposition \ref{CHprop}.
+Before diving into the details, we outline our strategy for the proof of Theorem \ref{thm:CH}.
 Let $p$ be a singular cell in $CH_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$.
 We say that $p\ot b$ is {\it localizable} if there exists $V \sub X$ such that
 \begin{itemize}
@@ -147,9 +147,7 @@
 $\supp(p)\cup\supp(b)$, and so on.
 
 
-\medskip
-
-\begin{proof}[Proof of Proposition \ref{CHprop}.]
+\begin{proof}[Proof of Theorem \ref{thm:CH}.]
 We'll use the notation $|b| = \supp(b)$ and $|p| = \supp(p)$.
 
 Choose a metric on $X$.
@@ -594,7 +592,7 @@
 	\gl:  R_*\ot CH_*(X, X) \otimes \bc_*(X)  \to R_*\ot CH_*(X\sgl, X \sgl) \otimes \bc_*(X \sgl) ,
 \]
 and it is easy to see that $\gl(G^m_*)\sub \ol{G}^m_*$.
-From this it follows that the diagram in the statement of Proposition \ref{CHprop} commutes.
+From this it follows that the diagram in the statement of Theorem \ref{thm:CH} commutes.
 
 \todo{this paragraph isn't very convincing, or at least I don't see what's going on}
 Finally we show that the action maps defined above are independent of
@@ -613,7 +611,7 @@
 Similar arguments show that this homotopy from $e$ to $e'$ is well-defined
 up to second order homotopy, and so on.
 
-This completes the proof of Proposition \ref{CHprop}.
+This completes the proof of Theorem \ref{thm:CH}.
 \end{proof}
 
 
@@ -629,7 +627,8 @@
 \end{rem*}
 
 
-\begin{prop}
+\begin{thm}
+\label{thm:CH-associativity}
 The $CH_*(X, Y)$ actions defined above are associative.
 That is, the following diagram commutes up to homotopy:
 \[ \xymatrix{
@@ -639,10 +638,10 @@
 } \]
 Here $\mu:CH_*(X, Y) \ot CH_*(Y, Z)\to CH_*(X, Z)$ is the map induced by composition
 of homeomorphisms.
-\end{prop}
+\end{thm}
 
 \begin{proof}
-The strategy of the proof is similar to that of Proposition \ref{CHprop}.
+The strategy of the proof is similar to that of Theorem \ref{thm:CH}.
 We will identify a subcomplex 
 \[
 	G_* \sub CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X)
@@ -656,7 +655,7 @@
 (If $p:P\times X\to Y$, then $p\inv(|q|)$ means the union over all $x\in P$ of 
 $p(x, \cdot)\inv(|q|)$.)
 
-As in the proof of Proposition \ref{CHprop}, we can construct a homotopy 
+As in the proof of Theorem \ref{thm:CH}, we can construct a homotopy 
 between the upper and lower maps restricted to $G_*$.
 This uses the facts that the maps agree on $CH_0(X, Y) \ot CH_0(Y, Z) \ot \bc_*(X)$,
 that they are compatible with gluing, and the contractibility of $\bc_*(X)$.

--- a/text/hochschild.tex	Mon Jul 12 17:29:25 2010 -0600
+++ b/text/hochschild.tex	Wed Jul 14 11:06:11 2010 -0600
@@ -19,7 +19,7 @@
 to find a more ``local" description of the Hochschild complex.
 
 Let $C$ be a *-1-category.
-Then specializing the definitions from above to the case $n=1$ we have:
+Then specializing the definition of the associated system of fields from \S \ref{sec:example:traditional-n-categories(fields)} above to the case $n=1$ we have:
 \begin{itemize}
 \item $\cC(pt) = \ob(C)$ .
 \item Let $R$ be a 1-manifold and $c \in \cC(\bd R)$.
@@ -44,8 +44,7 @@
 In order to prove this we will need to extend the 
 definition of the blob complex to allow points to also
 be labeled by elements of $C$-$C$-bimodules.
-(See Subsections \ref{moddecss} and \ref{ssec:spherecat} for a more general (i.e.\ $n>1$)
-version of this construction.)
+(See Subsections \ref{moddecss} and \ref{ssec:spherecat} for a more general version of this construction that applies in all dimensions.)
 
 Fix points $p_1, \ldots, p_k \in S^1$ and $C$-$C$-bimodules $M_1, \ldots M_k$.
 We define a blob-like complex $K_*(S^1, (p_i), (M_i))$.
@@ -79,8 +78,8 @@
 The complex $K_*(C)$ (here $C$ is being thought of as a
 $C$-$C$-bimodule, not a category) is homotopy equivalent to the blob complex
 $\bc_*(S^1; C)$.
-(Proof later.)
 \end{lem}
+The proof appears below.
 
 Next, we show that for any $C$-$C$-bimodule $M$,
 \begin{prop} \label{prop:hoch}
@@ -249,7 +248,7 @@
 \[
 	\bd j_\ep  + j_\ep \bd = \id - i \circ s .
 \]
-\nn{need to check signs coming from blob complex differential}
+(To get the signs correct here, we add $N_\ep$ as the first blob.)
 Since for $\ep$ small enough $L_*^\ep$ captures all of the
 homology of $\bc_*(S^1)$, 
 it follows that the mapping cone of $i \circ s$ is acyclic and therefore (using the fact that
@@ -288,11 +287,11 @@
 such that $\sum_i a_i q_i b_i = 0$ is in the image of $\ker(C \tensor E \tensor C \to C)$ under $\hat{g}$.
 For each $i$, we can find $\widetilde{q_i}$ so $g(\widetilde{q_i}) = q_i$.
 However $\sum_i a_i \widetilde{q_i} b_i$ need not be zero.
-Consider then $$\widetilde{q} = \sum_i (a_i \tensor \widetilde{q_i} \tensor b_i) - 1 \tensor (\sum_i a_i \widetilde{q_i} b_i) \tensor 1.$$ Certainly
+Consider then $$\widetilde{q} = \sum_i \left(a_i \tensor \widetilde{q_i} \tensor b_i\right) - 1 \tensor \left(\sum_i a_i \widetilde{q_i} b_i\right) \tensor 1.$$ Certainly
 $\widetilde{q} \in \ker(C \tensor E \tensor C \to E)$.
 Further,
 \begin{align*}
-\hat{g}(\widetilde{q}) & = \sum_i (a_i \tensor g(\widetilde{q_i}) \tensor b_i) - 1 \tensor (\sum_i a_i g(\widetilde{q_i}) b_i) \tensor 1 \\
+\hat{g}(\widetilde{q}) & = \sum_i \left(a_i \tensor g(\widetilde{q_i}) \tensor b_i\right) - 1 \tensor \left(\sum_i a_i g(\widetilde{q_i}\right) b_i) \tensor 1 \\
                        & = q - 0
 \end{align*}
 (here we used that $g$ is a map of $C$-$C$ bimodules, and that $\sum_i a_i q_i b_i = 0$).
@@ -420,13 +419,13 @@
 Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin N_\ep$,
 and have an additional blob $N_\ep$ with label $y_i - s_\ep(y_i)$.
 Define $j_\ep(x) = \sum x_i$.
-\nn{need to check signs coming from blob complex differential}
 Note that if $x \in K'_* \cap K_*^\ep$ then $j_\ep(x) \in K'_*$ also.
 
 The key property of $j_\ep$ is
 \eq{
     \bd j_\ep + j_\ep \bd = \id - \sigma_\ep.
 }
+(Again, to get the correct signs, $N_\ep$ must be added as the first blob.)
 If $j_\ep$ were defined on all of $K_*(C\otimes C)$, this would show that $\sigma_\ep$
 is a homotopy inverse to the inclusion $K'_* \to K_*(C\otimes C)$.
 One strategy would be to try to stitch together various $j_\ep$ for progressively smaller
@@ -531,12 +530,12 @@
 	\bd(m\otimes a)  & = & ma - am \\
 	\bd(m\otimes a \otimes b) & = & ma\otimes b - m\otimes ab + bm \otimes a .
 }
-In degree 0, we send $m\in M$ to the 0-blob diagram $\mathfig{0.05}{hochschild/0-chains}$; the base point
+In degree 0, we send $m\in M$ to the 0-blob diagram $\mathfig{0.04}{hochschild/0-chains}$; the base point
 in $S^1$ is labeled by $m$ and there are no other labeled points.
 In degree 1, we send $m\ot a$ to the sum of two 1-blob diagrams
 as shown in Figure \ref{fig:hochschild-1-chains}.
 
-\begin{figure}[t]
+\begin{figure}[ht]
 \begin{equation*}
 \mathfig{0.4}{hochschild/1-chains}
 \end{equation*}
@@ -547,19 +546,23 @@
 \label{fig:hochschild-1-chains}
 \end{figure}
 
-\begin{figure}[t]
+\begin{figure}[ht]
 \begin{equation*}
 \mathfig{0.6}{hochschild/2-chains-0}
 \end{equation*}
+\caption{The 0-chains in the image of $m \tensor a \tensor b$.}
+\label{fig:hochschild-2-chains-0}
+\end{figure}
+\begin{figure}[ht]
 \begin{equation*}
 \mathfig{0.4}{hochschild/2-chains-1} \qquad \mathfig{0.4}{hochschild/2-chains-2}
 \end{equation*}
-\caption{The 0-, 1- and 2-chains in the image of $m \tensor a \tensor b$.
-Only the supports of the 1- and 2-blobs are shown.}
-\label{fig:hochschild-2-chains}
+\caption{The 1- and 2-chains in the image of $m \tensor a \tensor b$.
+Only the supports of the blobs are shown, but see Figure \ref{fig:hochschild-example-2-cell} for an example of a $2$-cell label.}
+\label{fig:hochschild-2-chains-12}
 \end{figure}
 
-\begin{figure}[t]
+\begin{figure}[ht]
 \begin{equation*}
 A = \mathfig{0.1}{hochschild/v_1} + \mathfig{0.1}{hochschild/v_2} + \mathfig{0.1}{hochschild/v_3} + \mathfig{0.1}{hochschild/v_4}
 \end{equation*}
@@ -567,20 +570,20 @@
 v_1 & = \mathfig{0.05}{hochschild/v_1-1} -  \mathfig{0.05}{hochschild/v_1-2} &  v_2 & = \mathfig{0.05}{hochschild/v_2-1} -  \mathfig{0.05}{hochschild/v_2-2} \\ 
 v_3 & = \mathfig{0.05}{hochschild/v_3-1} -  \mathfig{0.05}{hochschild/v_3-2} &  v_4 & = \mathfig{0.05}{hochschild/v_4-1} -  \mathfig{0.05}{hochschild/v_4-2}
 \end{align*}
-\caption{One of the 2-cells from Figure \ref{fig:hochschild-2-chains}.}
+\caption{One of the 2-cells from Figure \ref{fig:hochschild-2-chains-12}.}
 \label{fig:hochschild-example-2-cell}
 \end{figure}
 
 In degree 2, we send $m\ot a \ot b$ to the sum of 24 ($=6\cdot4$) 2-blob diagrams as shown in
-Figure \ref{fig:hochschild-2-chains}.
-In Figure \ref{fig:hochschild-2-chains} the 1- and 2-blob diagrams are indicated only by their support.
+Figures \ref{fig:hochschild-2-chains-0} and \ref{fig:hochschild-2-chains-12}.
+In Figure \ref{fig:hochschild-2-chains-12} the 1- and 2-blob diagrams are indicated only by their support.
 We leave it to the reader to determine the labels of the 1-blob diagrams.
 Each 2-cell in the figure is labeled by a ball $V$ in $S^1$ which contains the support of all
 1-blob diagrams in its boundary.
 Such a 2-cell corresponds to a sum of the 2-blob diagrams obtained by adding $V$
 as an outer (non-twig) blob to each of the 1-blob diagrams in the boundary of the 2-cell.
 Figure \ref{fig:hochschild-example-2-cell} shows this explicitly for the 2-cell
-labeled $A$ in Figure \ref{fig:hochschild-2-chains}.
+labeled $A$ in Figure \ref{fig:hochschild-2-chains-12}.
 Note that the (blob complex) boundary of this sum of 2-blob diagrams is
 precisely the sum of the 1-blob diagrams corresponding to the boundary of the 2-cell.
 (Compare with the proof of \ref{bcontract}.)

--- a/text/intro.tex	Mon Jul 12 17:29:25 2010 -0600
+++ b/text/intro.tex	Wed Jul 14 11:06:11 2010 -0600
@@ -10,9 +10,9 @@
 \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 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 
+\item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category, 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 $M$.
+on the configuration space of unlabeled points in $M$. (See \S \ref{sec:comm_alg}.)
 %$$H_*(\bc_*(M; k[t])) = H^{\text{sing}}_*(\Delta^\infty(M), k).$$ 
 \end{itemize}
 The blob complex definition is motivated by the desire for a derived analogue of the usual TQFT Hilbert space 
@@ -111,13 +111,12 @@
 \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) 
+Section \S \ref{sec:deligne} gives
 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 the ``small blob complex", 
 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.
+as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. 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 $M$.
 
 
 \nn{some more things to cover in the intro}
@@ -150,13 +149,14 @@
 It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together
 with a link $L \subset \bd W$.
 The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$.
+\todo{I'm tempted to replace $A_{Kh}$ with $\cl{Kh}$ throughout this page -S}
 
 How would we go about computing $A_{Kh}(W^4, L)$?
-For $A_{Kh}(B^4, L)$, the main tool is the exact triangle (long exact sequence)
+For the Khovanov homology of a link in $S^3$ the main tool is the exact triangle (long exact sequence)
 relating resolutions of a crossing.
 Unfortunately, the exactness breaks if we glue $B^4$ to itself and attempt
 to compute $A_{Kh}(S^1\times B^3, L)$.
-According to the gluing theorem for TQFTs-via-fields, gluing along $B^3 \subset \bd B^4$
+According to the gluing theorem for TQFTs, gluing along $B^3 \subset \bd B^4$
 corresponds to taking a coend (self tensor product) over the cylinder category
 associated to $B^3$ (with appropriate boundary conditions).
 The coend is not an exact functor, so the exactness of the triangle breaks.
@@ -201,7 +201,7 @@
 
 \subsection{Formal properties}
 \label{sec:properties}
-We now summarize the results of the paper in the following list of formal properties.
+The blob complex enjoys the following list of formal properties.
 
 \begin{property}[Functoriality]
 \label{property:functoriality}%
@@ -228,8 +228,8 @@
 \end{equation*}
 \end{property}
 
-If an $n$-manifold $X_\text{cut}$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$ submanifold of its boundary, 
-write $X_\text{glued} = X_\text{cut} \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$.
+If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$ submanifold of its boundary, 
+write $X_\text{gl} = X \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$.
 Note that this includes the case of gluing two disjoint manifolds together.
 \begin{property}[Gluing map]
 \label{property:gluing-map}%
@@ -237,10 +237,10 @@
 %\begin{equation*}
 %\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2).
 %\end{equation*}
-Given a gluing $X_\mathrm{cut} \to X_\mathrm{glued}$, there is
+Given a gluing $X \to X_\mathrm{gl}$, there is
 a natural map
 \[
-	\bc_*(X_\mathrm{cut}) \to \bc_*(X_\mathrm{glued}) 
+	\bc_*(X) \to \bc_*(X_\mathrm{gl}) 
 \]
 (natural with respect to homeomorphisms, and also associative with respect to iterated gluings).
 \end{property}
@@ -249,9 +249,9 @@
 \label{property:contractibility}%
 With field coefficients, the blob complex on an $n$-ball is contractible in the sense that it is homotopic to its $0$-th homology.
 Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces associated by the system of fields $\cC$ to balls.
-\begin{equation}
+\begin{equation*}
 \xymatrix{\bc_*^{\cC}(B^n) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*^{\cC}(B^n)) \ar[r]^(0.6)\iso & \cC(B^n)}
-\end{equation}
+\end{equation*}
 \end{property}
 
 Properties \ref{property:functoriality} will be immediate from the definition given in
@@ -263,8 +263,9 @@
 
 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}%
+\newtheorem*{thm:skein-modules}{Theorem \ref{thm:skein-modules}}
+
+\begin{thm:skein-modules}[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 $\cC$.
@@ -272,7 +273,7 @@
 \begin{equation*}
 H_0(\bc_*^{\cC}(X)) \iso A^{\cC}(X)
 \end{equation*}
-\end{thm}
+\end{thm:skein-modules}
 
 \newtheorem*{thm:hochschild}{Theorem \ref{thm:hochschild}}
 
@@ -286,22 +287,25 @@
 
 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.
+We also note \S \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{thm}[$C_*(\Homeo(-))$ action]\mbox{}\\
+
+\newtheorem*{thm:CH}{Theorem \ref{thm:CH}}
+
+\begin{thm:CH}[$C_*(\Homeo(-))$ action]\mbox{}\\
 \vspace{-0.5cm}
 \label{thm:evaluation}%
-\begin{enumerate}
-\item There is a chain map
+There is a chain map
 \begin{equation*}
 \ev_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X).
 \end{equation*}
-
+such that
+\begin{enumerate}
 \item Restricted to $C_0(\Homeo(X))$ this is the action of homeomorphisms described in Property \ref{property:functoriality}. 
 
 \item For
@@ -315,17 +319,24 @@
             \bc_*(X) \ar[u]_{\gl_Y}
 }
 \end{equation*}
-\item Any such chain map satisfying points 2. and 3. above is unique, up to an iterated homotopy.
+\end{enumerate}
+Moreover any such chain map 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).
+\end{thm:CH}
+
+\newtheorem*{thm:CH-associativity}{Theorem \ref{thm:CH-associativity}}
+
+
+Further,
+\begin{thm:CH-associativity}
+\item The chain map of Theorem \ref{thm:CH} 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{thm}
+\end{thm:CH-associativity}
 
 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps
 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$
@@ -336,7 +347,8 @@
 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{thm}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category]
+\todo{Give this a number inside the text}
+\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. 
@@ -351,10 +363,11 @@
 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.
 \end{rem}
+Theorem \ref{thm:blobs-ainfty} appears as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats}
 
 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category
 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, $\cl{\cC}(M)$.
+The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit.
 
 \newtheorem*{thm:product}{Theorem \ref{thm:product}}
 
@@ -381,17 +394,15 @@
 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an
 $A_\infty$ module for $\bc_*(Y)$.
 
-\item For any $n$-manifold $X_\text{glued} = X_\text{cut} \bigcup_Y \selfarrow$, the blob complex $\bc_*(X_\text{glued})$ is the $A_\infty$ self-tensor product of
-$\bc_*(X_\text{cut})$ as an $\bc_*(Y)$-bimodule:
+\item For any $n$-manifold $X_\text{gl} = X\bigcup_Y \selfarrow$, the blob complex $\bc_*(X_\text{gl})$ is the $A_\infty$ self-tensor product of
+$\bc_*(X)$ as an $\bc_*(Y)$-bimodule:
 \begin{equation*}
-\bc_*(X_\text{glued}) \simeq \bc_*(X_\text{cut}) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow
+\bc_*(X_\text{gl}) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow
 \end{equation*}
 \end{itemize}
 \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}.
+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}.
 
 \subsection{Applications}
 \label{sec:applications}
@@ -426,7 +437,7 @@
 We could presumably also replace many of our chain complexes with topological spaces (or indeed, work at the generality of model categories), 
 and likely it will prove useful to think about the connections between what we do here and $(\infty,k)$-categories.
 More could be said about finite characteristic 
-(there appears in be $2$-torsion in $\bc_1(S^2, \cC)$ for any spherical $2$-category $\cC$, for example).
+(there appears in be $2$-torsion in $\bc_1(S^2; \cC)$ for any spherical $2$-category $\cC$, for example).
 Much more could be said about other types of manifolds, in particular oriented, 
 $\operatorname{Spin}$ and $\operatorname{Pin}^{\pm}$ manifolds, where boundary issues become more complicated.
 (We'd recommend thinking about boundaries as germs, rather than just codimension $1$ manifolds.) 
@@ -438,10 +449,10 @@
 
 Many results in Hochschild homology can be understood ``topologically" via the blob complex.
 For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$ 
-(see \cite[\S 4.2]{MR1600246}) simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1], A)$, 
+(see \cite[\S 4.2]{MR1600246}) simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1]; A)$, 
 but haven't investigated the details.
 
-Most importantly, however, \nn{applications!} \nn{cyclic homology, $n=2$ cases, contact, Kh}
+Most importantly, however, \nn{applications!} \nn{cyclic homology, $n=2$ cases, contact, Kh} \nn{stabilization} \nn{stable categories, generalized cohomology theories}
 
 
 \subsection{Thanks and acknowledgements}

--- a/text/ncat.tex	Mon Jul 12 17:29:25 2010 -0600
+++ b/text/ncat.tex	Wed Jul 14 11:06:11 2010 -0600
@@ -619,7 +619,7 @@
 \]
 These action maps are required to be associative up to homotopy
 \nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that
-a diagram like the one in Proposition \ref{CHprop} commutes.
+a diagram like the one in Theorem \ref{thm:CH} commutes.
 \nn{repeat diagram here?}
 \nn{restate this with $\Homeo(X\to X')$?  what about boundary fixing property?}
 \end{axiom}
@@ -1371,9 +1371,9 @@
 \]
 Here $C_*$ means singular chains and $\Homeo_\bd(M)$ is the space of homeomorphisms of $M$
 which fix $\bd M$.
-These action maps are required to be associative up to homotopy, 
+These action maps are required to be associative up to homotopy, as in Theorem \ref{thm:CH-associativity}, 
 and also compatible with composition (gluing) in the sense that
-a diagram like the one in Proposition \ref{CHprop} commutes.
+a diagram like the one in Theorem \ref{thm:CH} commutes.
 \end{module-axiom}
 
 As with the $n$-category version of the above axiom, we should also have families of collar maps act.

--- a/text/tqftreview.tex	Mon Jul 12 17:29:25 2010 -0600
+++ b/text/tqftreview.tex	Wed Jul 14 11:06:11 2010 -0600
@@ -245,8 +245,9 @@
 One of the advantages of string diagrams over pasting diagrams is that one has more
 flexibility in slicing them up in various ways.
 In addition, string diagrams are traditional in quantum topology.
-The diagrams predate by many years the terms ``string diagram" and ``quantum topology".
-\nn{?? cite penrose, kauffman, jones(?)}
+The diagrams predate by many years the terms ``string diagram" and ``quantum topology", e.g. \cite{
+MR0281657,MR776784 % penrose
+}
 
 If $X$ has boundary, we require that the cell decompositions are in general
 position with respect to the boundary --- the boundary intersects each cell
@@ -315,7 +316,7 @@
 
 In addition, we regard the labelings as being equivariant with respect to the * structure
 on 1-morphisms and pivotal structure on 2-morphisms.
-That is, we mod out my the relation which flips the transverse orientation of a 1-cell 
+That is, we mod out by the relation which flips the transverse orientation of a 1-cell 
 and replaces its label $a$ by $a^*$, as well as the relation which changes the parameterization of the link
 of a 0-cell and replaces its label by the appropriate pivotal conjugate.
 
@@ -378,12 +379,10 @@
 In this subsection we briefly review the construction of a TQFT from a system of fields and local relations.
 As usual, see \cite{kw:tqft} for more details.
 
-Let $W$ be an $n{+}1$-manifold.
-We can think of the path integral $Z(W)$ as assigning to each
+We can think of a path integral $Z(W)$ of an $n+1$-manifold (which we're not defining in this context; this is just motivation) as assigning to each
 boundary condition $x\in \cC(\bd W)$ a complex number $Z(W)(x)$.
 In other words, $Z(W)$ lies in $\c^{\lf(\bd W)}$, the vector space of linear
 maps $\lf(\bd W)\to \c$.
-(We haven't defined a path integral in this context; this is just for motivation.)
 
 The locality of the TQFT implies that $Z(W)$ in fact lies in a subspace
 $Z(\bd W) \sub \c^{\lf(\bd W)}$ defined by local projections.
@@ -400,12 +399,12 @@
 \label{defn:TQFT-invariant}
 The TQFT invariant of $X$ associated to a system of fields $\cF$ and local relations $\cU$ is 
 	$$A(X) \deq \lf(X) / U(X),$$
-where $\cU(X) \sub \lf(X)$ to be the space of local relations in $\lf(X)$;
-$\cU(X)$ is generated by things of the form $u\bullet r$, where
+where $\cU(X) \sub \lf(X)$ is the space of local relations in $\lf(X)$:
+$\cU(X)$ is generated by fields of the form $u\bullet r$, where
 $u\in U(B)$ for some embedded $n$-ball $B\sub X$ and $r\in \cC(X\setmin B)$.
 \end{defn}
-(The blob complex, defined in the next section, 
-is in some sense the derived version of $A(X)$.)
+The blob complex, defined in the next section, 
+is in some sense the derived version of $A(X)$.
 If $X$ has boundary we can similarly define $A(X; c)$ for each 
 boundary condition $c\in\cC(\bd X)$.
 
@@ -413,28 +412,28 @@
 a $k$-category $A(Y)$ to an $n{-}k$-manifold $Y$, for $0 \le k \le n$.
 These invariants fit together via actions and gluing formulas.
 We describe only the case $k=1$ below.
-(The construction of the $n{+}1$-dimensional part of the theory (the path integral) 
+The construction of the $n{+}1$-dimensional part of the theory (the path integral) 
 requires that the starting data (fields and local relations) satisfy additional
 conditions.
 We do not assume these conditions here, so when we say ``TQFT" we mean a decapitated TQFT
-that lacks its $n{+}1$-dimensional part.)
+that lacks its $n{+}1$-dimensional part. Such a ``decapitated'' TQFT is sometimes also called an $n+\epsilon$ or $n+\frac{1}{2}$ dimensional TQFT, referring to the fact that it assigns maps to mapping cylinders between $n$-manifolds, but nothing to arbitrary $n{+}1$-manifolds.
 
 Let $Y$ be an $n{-}1$-manifold.
-Define a (linear) 1-category $A(Y)$ as follows.
-The objects of $A(Y)$ are $\cC(Y)$.
+Define a linear 1-category $A(Y)$ as follows.
+The set of objects of $A(Y)$ is $\cC(Y)$.
 The morphisms from $a$ to $b$ are $A(Y\times I; a, b)$, 
 where $a$ and $b$ label the two boundary components of the cylinder $Y\times I$.
 Composition is given by gluing of cylinders.
 
 Let $X$ be an $n$-manifold with boundary and consider the collection of vector spaces
-$A(X; \cdot) \deq \{A(X; c)\}$ where $c$ ranges through $\cC(\bd X)$.
+$A(X; -) \deq \{A(X; c)\}$ where $c$ ranges through $\cC(\bd X)$.
 This collection of vector spaces affords a representation of the category $A(\bd X)$, where
 the action is given by gluing a collar $\bd X\times I$ to $X$.
 
 Given a splitting $X = X_1 \cup_Y X_2$ of a closed $n$-manifold $X$ along an $n{-}1$-manifold $Y$,
-we have left and right actions of $A(Y)$ on $A(X_1; \cdot)$ and $A(X_2; \cdot)$.
+we have left and right actions of $A(Y)$ on $A(X_1; -)$ and $A(X_2; -)$.
 The gluing theorem for $n$-manifolds states that there is a natural isomorphism
 \[
-	A(X) \cong A(X_1; \cdot) \otimes_{A(Y)} A(X_2; \cdot) .
+	A(X) \cong A(X_1; -) \otimes_{A(Y)} A(X_2; -) .
 \]
-
+A proof of this gluing formula appears in \cite{kw:tqft}, but it also becomes a special case of Theorem \ref{thm:gluing} by taking $0$-th homology.