--- a/text/a_inf_blob.tex Fri Jun 25 09:48:24 2010 -0700
+++ b/text/a_inf_blob.tex Sat Jun 26 16:31:28 2010 -0700
@@ -42,7 +42,7 @@
\nn{need to settle on notation; proof and statement are inconsistent}
-\begin{thm} \label{product_thm}
+\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 $C^{\times F}$ defined by
\begin{equation*}
@@ -57,7 +57,7 @@
\end{thm}
-\begin{proof}%[Proof of Theorem \ref{product_thm}]
+\begin{proof}
We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}.
First we define a map
@@ -214,7 +214,7 @@
collection of acyclic subcomplexes, so by the usual MoAM argument these two maps
are homotopic.
-This concludes the proof of Theorem \ref{product_thm}.
+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}
@@ -227,12 +227,12 @@
The new-fangled and old-fashioned blob complexes are homotopic.
\end{cor}
\begin{proof}
-Apply Theorem \ref{product_thm} with the fiber $F$ equal to a point.
+Apply Theorem \ref{thm:product} with the fiber $F$ equal to a point.
\end{proof}
\medskip
-Theorem \ref{product_thm} extends to the case of general fiber bundles
+Theorem \ref{thm:product} extends to the case of general fiber bundles
\[
F \to E \to Y .
\]
@@ -247,7 +247,7 @@
Let $\cF_E$ denote this $k$-category over $Y$.
We can adapt the homotopy colimit construction (based decompositions of $Y$ into balls) to
get a chain complex $\cF_E(Y)$.
-The proof of Theorem \ref{product_thm} goes through essentially unchanged
+The proof of Theorem \ref{thm:product} goes through essentially unchanged
to show that
\[
\bc_*(E) \simeq \cF_E(Y) .
@@ -298,9 +298,9 @@
\begin{proof}
\nn{for now, just prove $k=0$ case.}
-The proof is similar to that of Theorem \ref{product_thm}.
+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{product_thm}.
+the proof of Theorem \ref{thm:product}.
Let $\cT$ denote the chain complex $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.
Recall that this is a homotopy colimit based on decompositions of the interval $J$.
@@ -316,17 +316,15 @@
a subcomplex of $G_*$.
Next we define a map $\phi:G_*\to \cT$ using the method of acyclic models.
-As in the proof of Theorem \ref{product_thm}, we assign to a generator $a$ of $G_*$
+As in the proof of Theorem \ref{thm:product}, we assign to a generator $a$ of $G_*$
an acyclic subcomplex which is (roughly) $\psi\inv(a)$.
The proof of acyclicity is easier in this case since any pair of decompositions of $J$ have
a common refinement.
The proof that these two maps are inverse to each other is the same as in
-Theorem \ref{product_thm}.
+Theorem \ref{thm:product}.
\end{proof}
-This establishes Property \ref{property:gluing}.
-
\noop{
Let $\cT$ denote the $n{-}k$-category $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.
Let $D$ be an $n{-}k$-ball.
@@ -337,13 +335,14 @@
on $D\times X$, a decomposition of $J$ such that $b$ splits on the corresponding
decomposition of $D\times X$.
The proof that these two maps are inverse to each other is the same as in
-Theorem \ref{product_thm}.
+Theorem \ref{thm:product}.
}
\medskip
\subsection{Reconstructing mapping spaces}
+\label{sec:map-recon}
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,
@@ -353,7 +352,8 @@
want to know about spaces of maps of $k$-balls into $T$ ($k\le n$).
To simplify notation, let $\cT = \pi^\infty_{\leq n}(T)$.
-\begin{thm} \label{thm:map-recon}
+\begin{thm}
+\label{thm:map-recon}
The blob complex for $M$ with coefficients in the fundamental $A_\infty$ $n$-category for $T$
is quasi-isomorphic to singular chains on maps from $M$ to $T$.
$$\cB^\cT(M) \simeq C_*(\Maps(M\to T)).$$
@@ -369,7 +369,7 @@
\end{rem}
\begin{proof}
-The proof is again similar to that of Theorem \ref{product_thm}.
+The proof is again similar to that of Theorem \ref{thm:product}.
We begin by constructing chain map $\psi: \cB^\cT(M) \to C_*(\Maps(M\to T))$.
@@ -411,7 +411,7 @@
Define $D(a)$ to be the subcomplex of $\cB^\cT(M)$ generated by all
pairs $(b, \ol{K})$, where $b$ is a generator appearing in an iterated boundary of $a$
and $\ol{K}$ is an index of the homotopy colimit $\cB^\cT(M)$.
-(See the proof of Theorem \ref{product_thm} for more details.)
+(See the proof of Theorem \ref{thm:product} for more details.)
The same proof as of Lemma \ref{lem:d-a-acyclic} shows that $D(a)$ is acyclic.
By the usual acyclic models nonsense, there is a (unique up to homotopy)
map $\phi:G_*\to \cB^\cT(M)$ such that $\phi(a)\in D(a)$.
@@ -423,7 +423,7 @@
It is now easy to see that $\psi\circ\phi$ is the identity on the nose.
Another acyclic models argument shows that $\phi\circ\psi$ is homotopic to the identity.
-(See the proof of Theorem \ref{product_thm} for more details.)
+(See the proof of Theorem \ref{thm:product} for more details.)
\end{proof}
\noop{