
diff -r a02a6158f3bd -r a8b8ebcf07ac text/a_inf_blob.tex
--- a/text/a_inf_blob.tex	Sat Jun 26 16:31:28 2010 -0700
+++ b/text/a_inf_blob.tex	Sat Jun 26 17:22:53 2010 -0700
@@ -2,18 +2,13 @@
 
 \section{The blob complex for $A_\infty$ $n$-categories}
 \label{sec:ainfblob}
+Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we make the anticlimactically tautological definition of the blob
+complex $\bc_*(M;\cC)$ to be the homotopy colimit $\cl{\cC}(M)$ of Section \ref{ss:ncat_fields}.
 
-Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we define the blob
-complex $\bc_*(M)$ to the be the homotopy colimit $\cC(M)$ of Section \ref{sec:ncats}.
-\nn{say something about this being anticlimatically tautological?}
 We will show below 
 in Corollary \ref{cor:new-old}
-that this agrees (up to homotopy) with our original definition of the blob complex
-in the case of plain $n$-categories.
-When we need to distinguish between the new and old definitions, we will refer to the 
-new-fangled and old-fashioned blob complex.
-
-\medskip
+that when $\cC$ is obtained from a topological $n$-category $\cD$ as the blob complex of a point, this agrees (up to homotopy) with our original definition of the blob complex
+for $\cD$.
 
 An important technical tool in the proofs of this section is provided by the idea of `small blobs'.
 Fix $\cU$, an open cover of $M$.
@@ -44,15 +39,15 @@
 
 \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
+Example \ref{ex:blob-complexes-of-balls} that there is an  $A_\infty$ $k$-category $\bc_*(F; C)$ defined by
 \begin{equation*}
-C^{\times F}(B) = \cB_*(B \times F, C).
+\bc_*(F; C) = \cB_*(B \times F, C).
 \end{equation*}
 Now, given a $k$-manifold $Y$, there is a homotopy equivalence between the `old-fashioned' 
 blob complex for $Y \times F$ with coefficients in $C$ and the `new-fangled' 
-(i.e.\ homotopy colimit) blob complex for $Y$ with coefficients in $C^{\times F}$:
+(i.e.\ homotopy colimit) blob complex for $Y$ with coefficients in $\bc_*(F; C)$:
 \begin{align*}
-\cB_*(Y \times F, C) & \htpy \cB_*(Y, C^{\times F})
+\cB_*(Y \times F; C) & \htpy \cl{\bc_*(F; C)}(Y)
 \end{align*}
 \end{thm}
 
@@ -62,7 +57,7 @@
 
 First we define a map 
 \[
-	\psi: \bc_*^\cF(Y) \to \bc_*^C(Y\times F) .
+	\psi: \cl{\bc_*(F; C)}(Y) \to \bc_*(Y\times F;C) .
 \]
 In filtration degree 0 we just glue together the various blob diagrams on $X_i\times F$
 (where $X_i$ is a component of a permissible decomposition of $Y$) to get a blob diagram on
@@ -70,25 +65,25 @@
 In filtration degrees 1 and higher we define the map to be zero.
 It is easy to check that this is a chain map.
 
-In the other direction, we will define a subcomplex $G_*\sub \bc_*^C(Y\times F)$
+In the other direction, we will define a subcomplex $G_*\sub \bc_*(Y\times F;C)$
 and a map
 \[
-	\phi: G_* \to \bc_*^\cF(Y) .
+	\phi: G_* \to \cl{\bc_*(F; C)}(Y) .
 \]
 
 Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding
 decomposition of $Y\times F$ into the pieces $X_i\times F$.
 
-Let $G_*\sub \bc_*^C(Y\times F)$ be the subcomplex generated by blob diagrams $a$ such that there
+Let $G_*\sub \bc_*(Y\times F;C)$ be the subcomplex generated by blob diagrams $a$ such that there
 exists a decomposition $K$ of $Y$ such that $a$ splits along $K\times F$.
-It follows from Proposition \ref{thm:small-blobs} that $\bc_*^C(Y\times F)$ is homotopic to a subcomplex of $G_*$.
+It follows from Proposition \ref{thm:small-blobs} that $\bc_*(Y\times F; C)$ is homotopic to a subcomplex of $G_*$.
 (If the blobs of $a$ are small with respect to a sufficiently fine cover then their
 projections to $Y$ are contained in some disjoint union of balls.)
 Note that the image of $\psi$ is equal to $G_*$.
 
-We will define $\phi: G_* \to \bc_*^\cF(Y)$ using the method of acyclic models.
+We will define $\phi: G_* \to \cl{\bc_*(F; C)}(Y)$ using the method of acyclic models.
 Let $a$ be a generator of $G_*$.
-Let $D(a)$ denote the subcomplex of $\bc_*^\cF(Y)$ generated by all $(b, \ol{K})$
+Let $D(a)$ denote the subcomplex of $\cl{\bc_*(F; C)}(Y)$ generated by all $(b, \ol{K})$
 such that $a$ splits along $K_0\times F$ and $b$ is a generator appearing
 in an iterated boundary of $a$ (this includes $a$ itself).
 (Recall that $\ol{K} = (K_0,\ldots,K_l)$ denotes a chain of decompositions;
@@ -194,13 +189,13 @@
 \end{proof}
 
 We are now in a position to apply the method of acyclic models to get a map
-$\phi:G_* \to \bc_*^\cF(Y)$.
+$\phi:G_* \to \cl{\bc_*(F; C)}(Y)$.
 We may assume that $\phi(a)$ has the form $(a, K) + r$, where $(a, K)$ is in filtration degree zero
 and $r$ has filtration degree greater than zero.
 
 We now show that $\phi\circ\psi$ and $\psi\circ\phi$ are homotopic to the identity.
 
-$\psi\circ\phi$ is the identity on the nose:
+First, $\psi\circ\phi$ is the identity on the nose:
 \[
 	\psi(\phi(a)) = \psi((a,K)) + \psi(r) = a + 0.
 \]
@@ -208,10 +203,10 @@
 $\psi$ glues those pieces back together, yielding $a$.
 We have $\psi(r) = 0$ since $\psi$ is zero in positive filtration degrees.
  
-$\phi\circ\psi$ is the identity up to homotopy by another MoAM argument.
+Second, $\phi\circ\psi$ is the identity up to homotopy by another argument based on the method of acyclic models.
 To each generator $(b, \ol{K})$ of $G_*$ we associate the acyclic subcomplex $D(b)$ defined above.
 Both the identity map and $\phi\circ\psi$ are compatible with this
-collection of acyclic subcomplexes, so by the usual MoAM argument these two maps
+collection of acyclic subcomplexes, so by the usual method of acyclic models argument these two maps
 are homotopic.
 
 This concludes the proof of Theorem \ref{thm:product}.
@@ -221,10 +216,10 @@
 
 \medskip
 
-\todo{rephrase this}
 \begin{cor}
 \label{cor:new-old}
-The new-fangled and old-fashioned blob complexes are homotopic.
+The blob complex of a manifold $M$ with coefficients in a topological $n$-category $\cC$ is homotopic to the homotopy colimit invariant of $M$ defined using the $A_\infty$ $n$-category obtained by applying the blob complex to a point:
+$$\bc_*(M; \cC) \htpy \cl{\bc_*(pt; \cC)}(M).$$ 
 \end{cor}
 \begin{proof}
 Apply Theorem \ref{thm:product} with the fiber $F$ equal to a point.




