--- a/text/a_inf_blob.tex Thu May 27 15:06:48 2010 -0700
+++ b/text/a_inf_blob.tex Thu May 27 20:09:47 2010 -0700
@@ -15,6 +15,12 @@
\medskip
+\subsection{The small blob complex}
+
+\input{text/smallblobs}
+
+\subsection{A product formula}
+
Let $M^n = Y^k\times F^{n-k}$.
Let $C$ be a plain $n$-category.
Let $\cF$ be the $A_\infty$ $k$-category which assigns to a $k$-ball
@@ -25,7 +31,7 @@
new-fangled blob complex $\bc_*^\cF(Y)$.
\end{thm}
-\input{text/smallblobs}
+
\begin{proof}[Proof of Theorem \ref{product_thm}]
We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}.
@@ -213,6 +219,9 @@
\medskip
+\subsection{A gluing theorem}
+\label{sec:gluing}
+
Next we prove a gluing theorem.
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
@@ -230,6 +239,7 @@
\end{itemize}
\begin{thm}
+\label{thm:gluing}
$\bc(X) \cong \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.
\end{thm}
@@ -254,6 +264,8 @@
\medskip
+\subsection{Reconstructing mapping spaces}
+
The next theorem shows how to reconstruct a mapping space from local data.
Let $T$ be a topological space, let $M$ be an $n$-manifold,
and recall the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$