definition-izing the blob complex for an A_infty cat, and stating assumptions more prominently in S7.2
Binary file RefereeReport.pdf has changed
--- a/text/a_inf_blob.tex Thu Oct 06 12:11:47 2011 -0700
+++ b/text/a_inf_blob.tex Thu Oct 06 12:20:35 2011 -0700
@@ -2,9 +2,13 @@
\section{The blob complex for \texorpdfstring{$A_\infty$}{A-infinity} \texorpdfstring{$n$}{n}-categories}
\label{sec:ainfblob}
-Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we make the
+Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we make the following
anticlimactically tautological definition of the blob
-complex $\bc_*(M;\cC)$ to be the homotopy colimit $\cl{\cC}(M)$ of \S\ref{ss:ncat_fields}.
+complex.
+\begin{defn}
+The blob complex
+ $\bc_*(M;\cC)$ of an $n$-manifold $n$ with coefficients in an $A_\infty$ $n$-category is the homotopy colimit $\cl{\cC}(M)$ of \S\ref{ss:ncat_fields}.
+\end{defn}
We will show below
in Corollary \ref{cor:new-old}
@@ -335,7 +339,7 @@
\subsection{A gluing theorem}
\label{sec:gluing}
-Next we prove a gluing theorem.
+Next we prove a gluing theorem. Throughout this section fix a particular $n$-dimensional system of fields $\cE$ and local relations. Each blob complex below is with respect to this $\cE$.
Let $X$ be a closed $k$-manifold with a splitting $X = X'_1\cup_Y X'_2$.
We will need an explicit collar on $Y$, so rewrite this as
$X = X_1\cup (Y\times J) \cup X_2$.
@@ -364,7 +368,7 @@
\begin{thm}
\label{thm:gluing}
-When $k=n$ above, $\bc(X)$ is homotopy equivalent to $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.
+Suppose $X$ is an $n$-manifold, and $X = X_1\cup (Y\times J) \cup X_2$ (i.e. just as with $k=n$ above). Then $\bc(X)$ is homotopy equivalent to the $A_\infty$ tensor product $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.
\end{thm}
\begin{proof}