diff -r 7afacaa87bdb -r ff867bfc8e9c text/a_inf_blob.tex --- 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 $\pi^\infty_{\leq n}(T)$