
diff -r 379e9a10c079 -r 45807ce15615 text/a_inf_blob.tex
--- a/text/a_inf_blob.tex	Wed Jul 14 11:08:12 2010 -0600
+++ b/text/a_inf_blob.tex	Sat Jul 17 20:57:35 2010 -0600
@@ -7,12 +7,16 @@
 
 We will show below 
 in Corollary \ref{cor:new-old}
-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 $\cD$.
+that when $\cC$ is obtained from a system of fields $\cD$ 
+as the blob complex of an $n$-ball (see Example \ref{ex:blob-complexes-of-balls}), 
+$\cl{\cC}(M)$ is homotopy equivalent to
+our original definition of the blob complex $\bc_*^\cD(M)$.
 
-An important technical tool in the proofs of this section is provided by the idea of `small blobs'.
+\medskip
+
+An important technical tool in the proofs of this section is provided by the idea of ``small blobs".
 Fix $\cU$, an open cover of $M$.
-Define the `small blob complex' $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ of all blob diagrams in which every blob is contained in some open set of $\cU$, and moreover each field labeling a region cut out by the blobs is splittable into fields on smaller regions, each of which is contained in some open set.
+Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ of all blob diagrams in which every blob is contained in some open set of $\cU$, and moreover each field labeling a region cut out by the blobs is splittable into fields on smaller regions, each of which is contained in some open set of $\cU$.
 
 \begin{thm}[Small blobs] \label{thm:small-blobs}
 The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence.
@@ -22,42 +26,27 @@
 \subsection{A product formula}
 \label{ss:product-formula}
 
-\noop{
-Let $Y$ be a $k$-manifold, $F$ be an $n{-}k$-manifold, and 
-\[
-	E = Y\times F .
-\]
-Let $\cC$ be an $n$-category.
-Let $\cF$ be the $k$-category of Example \ref{ex:blob-complexes-of-balls}, 
-\[
-	\cF(X) = \cC(X\times F)
-\]
-for $X$ an $m$-ball with $m\le k$.
-}
 
-\nn{need to settle on notation; proof and statement are inconsistent}
+Given a system of fields $\cE$ and a $n{-}k$-manifold $F$, recall from 
+Example \ref{ex:blob-complexes-of-balls} that there is an  $A_\infty$ $k$-category $\cC_F$ 
+defined by $\cC_F(X) = \cE(X\times F)$ if $\dim(X) < k$ and
+$\cC_F(X) = \bc_*^\cE(X\times F)$ if $\dim(X) = k$.
+
 
 \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 $\bc_*(F; C)$ defined by
-\begin{equation*}
-\bc_*(F; C)(B) = \cB_*(F \times B; 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 $\bc_*(F; C)$:
-\begin{align*}
-\cB_*(Y \times F; C) & \htpy \cl{\bc_*(F; C)}(Y)
-\end{align*}
-\end{thm}
-
+Let $Y$ be a $k$-manifold.
+Then there is a homotopy equivalence between ``old-fashioned" (blob diagrams) 
+and ``new-fangled" (hocolimit) blob complexes
+\[
+	\cB_*(Y \times F) \htpy \cl{\cC_F}(Y) .
+\]\end{thm}
 
 \begin{proof}
-We will use the concrete description of the colimit from \S\ref{ss:ncat_fields}.
+We will use the concrete description of the homotopy colimit from \S\ref{ss:ncat_fields}.
 
 First we define a map 
 \[
-	\psi: \cl{\bc_*(F; C)}(Y) \to \bc_*(Y\times F;C) .
+	\psi: \cl{\cC_F}(Y) \to \bc_*(Y\times F;C) .
 \]
 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
@@ -68,7 +57,7 @@
 In the other direction, we will define a subcomplex $G_*\sub \bc_*(Y\times F;C)$
 and a map
 \[
-	\phi: G_* \to \cl{\bc_*(F; C)}(Y) .
+	\phi: G_* \to \cl{\cC_F}(Y) .
 \]
 
 Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding
@@ -81,9 +70,9 @@
 projections to $Y$ are contained in some disjoint union of balls.)
 Note that the image of $\psi$ is equal to $G_*$.
 
-We will define $\phi: G_* \to \cl{\bc_*(F; C)}(Y)$ using the method of acyclic models.
+We will define $\phi: G_* \to \cl{\cC_F}(Y)$ using the method of acyclic models.
 Let $a$ be a generator of $G_*$.
-Let $D(a)$ denote the subcomplex of $\cl{\bc_*(F; C)}(Y)$ generated by all $(b, \ol{K})$
+Let $D(a)$ denote the subcomplex of $\cl{\cC_F}(Y)$ 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;
@@ -189,7 +178,7 @@
 \end{proof}
 
 We are now in a position to apply the method of acyclic models to get a map
-$\phi:G_* \to \cl{\bc_*(F; C)}(Y)$.
+$\phi:G_* \to \cl{\cC_F}(Y)$.
 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.
 




