# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1275541792 25200
# Node ID a20e2318cbb0e1c8c272613be2f632799a96148b
# Parent  6cc92b273d44ce18c99a484f5881e21a45c93080
rewrite proof from gluing thm

diff -r 6cc92b273d44 -r a20e2318cbb0 text/a_inf_blob.tex
--- a/text/a_inf_blob.tex	Wed Jun 02 16:51:40 2010 -0700
+++ b/text/a_inf_blob.tex	Wed Jun 02 22:09:52 2010 -0700
@@ -217,7 +217,7 @@
 This concludes the proof of Theorem \ref{product_thm}.
 \end{proof}
 
-\nn{need to say something about dim $< n$ above}
+\nn{need to prove a version where $E$ above has dimension $m<n$; result is an $n{-}m$-category}
 
 \medskip
 
@@ -238,8 +238,8 @@
 We outline one approach here and a second in Subsection xxxx.
 
 We can generalize the definition of a $k$-category by replacing the categories
-of $j$-balls ($j\le k$) with categories of $j$-balls $D$ equipped with a map $p:D\to Y$.
-\nn{need citation to other work that does this; Stolz and Teichner?}
+of $j$-balls ($j\le k$) with categories of $j$-balls $D$ equipped with a map $p:D\to Y$
+(c.f. \cite{MR2079378}).
 Call this a $k$-category over $Y$.
 A fiber bundle $F\to E\to Y$ gives an example of a $k$-category over $Y$:
 assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$.
@@ -276,27 +276,56 @@
 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
 $X = X_1\cup (Y\times J) \cup X_2$.
-Given this data we have: \nn{need refs to above for these}
+Given this data we have:
 \begin{itemize}
 \item An $A_\infty$ $n{-}k$-category $\bc(X)$, which assigns to an $m$-ball
 $D$ fields on $D\times X$ (for $m+k < n$) or the blob complex $\bc_*(D\times X; c)$
-(for $m+k = n$). \nn{need to explain $c$}.
+(for $m+k = n$).
+(See Example \ref{ex:blob-complexes-of-balls}.)
+%\nn{need to explain $c$}.
 \item An $A_\infty$ $n{-}k{+}1$-category $\bc(Y)$, defined similarly.
 \item Two $\bc(Y)$ modules $\bc(X_1)$ and $\bc(X_2)$, which assign to a marked
 $m$-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 \nn{need example for this}.)
 \end{itemize}
 
 \begin{thm}
 \label{thm:gluing}
-$\bc(X) \cong \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.
+$\bc(X) \simeq \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.
 \end{thm}
 
 \begin{proof}
+\nn{for now, just prove $k=0$ case.}
 The proof is similar to that of Theorem \ref{product_thm}.
-\nn{need to say something about dimensions less than $n$, 
-but for now concentrate on top dimension.}
+We give a short sketch with emphasis on the differences from 
+the proof of Theorem \ref{product_thm}.
+
+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 $J$.
+
+We define a map $\psi:\cT\to \bc_*(X)$.  On filtration degree zero summands it is given
+by gluing the pieces together to get a blob diagram on $X$.
+On filtration degree 1 and greater $\psi$ is zero.
 
+The image of $\psi$ is the subcomplex $G_*\sub \bc(X)$ generated by blob diagrams which split
+over some decomposition of $J$.
+It follows from Proposition \ref{thm:small-blobs} that $\bc_*(X)$ is homotopic to 
+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 $G_*$
+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}.
+\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.
 There is an obvious map from $\cT(D)$ to $\bc_*(D\times X)$.
@@ -307,9 +336,8 @@
 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}.
-\end{proof}
+}
 
-This establishes Property \ref{property:gluing}.
 
 \medskip
 
diff -r 6cc92b273d44 -r a20e2318cbb0 text/ncat.tex
--- a/text/ncat.tex	Wed Jun 02 16:51:40 2010 -0700
+++ b/text/ncat.tex	Wed Jun 02 22:09:52 2010 -0700
@@ -1044,7 +1044,7 @@
 Suppose $S$ is a topological space, with a subspace $T$. We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$ for $k<n$ the set $\pi_{\leq n}(S,T)(B,N)$ consists of all continuous maps of pairs $(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all such maps modulo homotopies fixed on $\bdy B \setminus N$. This is a module over the fundamental $n$-category $\pi_{\leq n}(S)$ of $S$, from Example \ref{ex:maps-to-a-space}. Modifications corresponding to Examples \ref{ex:maps-to-a-space-with-a-fiber} and \ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains.
 \end{example}
 
-\subsection{Modules as boundary labels}
+\subsection{Modules as boundary labels (colimits for decorated manifolds)}
 \label{moddecss}
 
 Fix a topological $n$-category or $A_\infty$ $n$-category  $\cC$. Let $W$ be a $k$-manifold ($k\le n$),