--- a/text/a_inf_blob.tex Wed Sep 22 19:01:41 2010 -0700
+++ b/text/a_inf_blob.tex Wed Sep 22 20:42:47 2010 -0700
@@ -31,7 +31,7 @@
\label{ss:product-formula}
-Given a system of fields $\cE$ and a $n{-}k$-manifold $F$, recall from
+Given an $n$-dimensional 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
$\cC_F(X) = \bc_*(X\times F;\cE)$ if $\dim(X) = k$.
@@ -200,11 +200,21 @@
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}
+%\nn{need to prove a version where $E$ above has dimension $m<n$; result is an $n{-}m$-category}
+
+If $Y$ has dimension $k-m$, then we have an $m$-category $\cC_{Y\times F}$ whose value at
+a $j$-ball $X$ is either $\cE(X\times Y\times F)$ (if $j<m$) or $\bc_*(X\times Y\times F)$
+(if $j=m$).
+(See Example \ref{ex:blob-complexes-of-balls}.)
+Similarly we have an $m$-category whose value at $X$ is $\cl{\cC_F}(X\times Y)$.
+These two categories are equivalent, but since we do not define functors between
+topological $n$-categories in this paper we are unable to say precisely
+what ``equivalent" means in this context.
+We hope to include this stronger result in a future paper.
\medskip
-Taking $F$ above to be a point, we obtain the following corollary.
+Taking $F$ in Theorem \ref{thm:product} to be a point, we obtain the following corollary.
\begin{cor}
\label{cor:new-old}
@@ -324,18 +334,24 @@
$m$-ball $(D, H)$ either fields on $(D\times Y) \cup (H\times X_i)$ (if $m+k < n$)
or the blob complex $\bc_*((D\times Y) \cup (H\times X_i))$ (if $m+k = n$).
(See Example \ref{bc-module-example}.)
+\item The tensor product $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$, which is
+an $A_\infty$ $n{-}k$-category.
+(See \S \ref{moddecss}.)
\end{itemize}
-\nn{statement (and proof) is only for case $k=n$; need to revise either above or below; maybe
-just say that until we define functors we can't do more}
+It is the case that the $n{-}k$-categories $\bc(X)$ and $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$
+are equivalent for all $k$, but since we do not develop a definition of functor between $n$-categories
+in this paper, we cannot state this precisely.
+(It will appear in a future paper.)
+So we content ourselves with
\begin{thm}
\label{thm:gluing}
-$\bc(X) \simeq \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.
+When $k=n$ above, $\bc(X)$ is homotopy equivalent to $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.
\end{thm}
\begin{proof}
-We will assume $k=n$; the other cases are similar.
+%We will assume $k=n$; the other cases are similar.
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{thm:product}.