# HG changeset patch
# User Scott Morrison <scott@tqft.net>
# Date 1275583638 25200
# Node ID bc22926d4fb0d8ef0be877693e54a13c7e95c955
# Parent  6ef67f13b69c01da0c4bedb8c3ac9099d0992c27# Parent  d163ad9543a5e7970d1f0a6df8958606d6c85fd3
Automated merge with https://tqft.net/hg/blob/

diff -r d163ad9543a5 -r bc22926d4fb0 blob1.tex
--- a/blob1.tex	Wed Jun 02 17:45:13 2010 -0700
+++ b/blob1.tex	Thu Jun 03 09:47:18 2010 -0700
@@ -68,8 +68,6 @@
 
 \input{text/a_inf_blob}
 
-\input{text/comm_alg}
-
 \input{text/deligne}
 
 \appendix
@@ -80,6 +78,8 @@
 
 \input{text/appendixes/comparing_defs}
 
+\input{text/comm_alg}
+
 % ----------------------------------------------------------------
 %\newcommand{\urlprefix}{}
 \bibliographystyle{plain}
@@ -94,4 +94,3 @@
 \end{document}
 % ----------------------------------------------------------------
 
-
diff -r d163ad9543a5 -r bc22926d4fb0 preamble.tex
--- a/preamble.tex	Wed Jun 02 17:45:13 2010 -0700
+++ b/preamble.tex	Thu Jun 03 09:47:18 2010 -0700
@@ -190,6 +190,8 @@
 \newcommand{\CD}[1]{C_*(\Diff(#1))}
 \newcommand{\CH}[1]{C_*(\Homeo(#1))}
 
+\newcommand{\cl}[1]{\underrightarrow{#1}}
+
 \newcommand{\directSumStack}[2]{{\begin{matrix}#1 \\ \DirectSum \\#2\end{matrix}}}
 \newcommand{\directSumStackThree}[3]{{\begin{matrix}#1 \\ \DirectSum \\#2 \\ \DirectSum \\#3\end{matrix}}}
 
diff -r d163ad9543a5 -r bc22926d4fb0 text/a_inf_blob.tex
--- a/text/a_inf_blob.tex	Wed Jun 02 17:45:13 2010 -0700
+++ b/text/a_inf_blob.tex	Thu Jun 03 09:47:18 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
 
@@ -235,11 +235,11 @@
 \[
 	F \to E \to Y .
 \]
-We outline two approaches.
+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 $\bc_*(p^*(E))$.
@@ -254,6 +254,7 @@
 
 
 
+\nn{put this later}
 
 \nn{The second approach: Choose a decomposition $Y = \cup X_i$
 such that the restriction of $E$ to $X_i$ is a product $F\times X_i$.
@@ -275,28 +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
 $X = X_1\cup (Y\times J) \cup X_2$.
-\nn{need figure}
-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
 
@@ -329,12 +357,15 @@
 \end{thm}
 \begin{rem}
 \nn{This just isn't true, Lurie doesn't do this! I just heard this from Ricardo...}
+\nn{KW: Are you sure about that?}
 Lurie has shown in \cite{0911.0018} that the topological chiral homology of an $n$-manifold $M$ with coefficients in \nn{a certain $E_n$ algebra constructed from $T$} recovers  the same space of singular chains on maps from $M$ to $T$, with the additional hypothesis that $T$ is $n-1$-connected. This extra hypothesis is not surprising, in view of the idea that an $E_n$ algebra is roughly equivalent data as an $A_\infty$ $n$-category which is trivial at all but the topmost level.
 \end{rem}
 
+\nn{proof is again similar to that of Theorem \ref{product_thm}.  should probably say that explicitly}
+
 \begin{proof}
 We begin by constructing chain map $g: \cB^\cT(M) \to C_*(\Maps(M\to T))$.
-We then use \ref{extension_lemma_b} to show that $g$ induces isomorphisms on homology.
+We then use Lemma \ref{extension_lemma_c} to show that $g$ induces isomorphisms on homology.
 
 Recall that the homotopy colimit $\cB^\cT(M)$ is constructed out of a series of
 $j$-fold mapping cylinders, $j \ge 0$.
@@ -364,42 +395,12 @@
 It is not hard to see that this defines a chain map from 
 $\cB^\cT(M)$ to $C_*(\Maps(M\to T))$.
 
-
-%%%%%%%%%%%%%%%%%
-\noop{
-Next we show that $g$ induces a surjection on homology.
-Fix $k > 0$ and choose an open cover $\cU$ of $M$ fine enough so that the union 
-of any $k$ open sets of $\cU$ is contained in a disjoint union of balls in $M$.
-\nn{maybe should refer to elsewhere in this paper where we made a very similar argument}
-Let $S_*$ be the subcomplex of $C_*(\Maps(M\to T))$ generated by chains adapted to $\cU$.
-It follows from Lemma \ref{extension_lemma_b} that $C_*(\Maps(M\to T))$
-retracts onto $S_*$.
-
-Let $S_{\le k}$ denote the chains of $S_*$ of degree less than or equal to $k$.
-We claim that $S_{\le k}$ lies in the image of $g$.
-Let $c$ be a generator of $S_{\le k}$ --- that is, a $j$-parameter family of maps $M\to T$,
-$j \le k$.
-We chose $\cU$ fine enough so that the support of $c$ is contained in a disjoint union of balls
-in $M$.
-It follow that we can choose a decomposition $K$ of $M$ so that the support of $c$ is 
-disjoint from the $n{-}1$-skeleton of $K$.
-It is now easy to see that $c$ is in the image of $g$.
-
-Next we show that $g$ is injective on homology.
-}
-
-
-
 \nn{...}
 
-
-
 \end{proof}
 
 \nn{maybe should also mention version where we enrich over
-spaces rather than chain complexes; should comment on Lurie's \cite{0911.0018} (and others') similar result
-for the $E_\infty$ case, and mention that our version does not require 
-any connectivity assumptions}
+spaces rather than chain complexes;}
 
 \medskip
 \hrule
@@ -407,7 +408,7 @@
 
 \nn{to be continued...}
 \medskip
-\nn{still to do: fiber bundles, general maps}
+\nn{still to do: general maps}
 
 \todo{}
 Various citations we might want to make:
@@ -418,21 +419,4 @@
 \item \cite{MR1256989} definition of framed little-discs operad
 \end{itemize}
 
-We now turn to establishing the gluing formula for blob homology, restated from Property \ref{property:gluing} in the Introduction
-\begin{itemize}
-%\mbox{}% <-- gets the indenting right
-\item For any $(n-1)$-manifold $Y$, the blob homology of $Y \times I$ is
-naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below.
 
-\item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob homology of $X$ is naturally an
-$A_\infty$ module for $\bc_*(Y \times I)$.
-
-\item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension
-$0$-submanifold of its boundary, the blob homology of $X'$, obtained from
-$X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of
-$\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule.
-\begin{equation*}
-\bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \!\!\!\!\!\!\xymatrix{ \ar@(ru,rd)@<-1ex>[]}
-\end{equation*}
-\end{itemize}
-
diff -r d163ad9543a5 -r bc22926d4fb0 text/appendixes/famodiff.tex
--- a/text/appendixes/famodiff.tex	Wed Jun 02 17:45:13 2010 -0700
+++ b/text/appendixes/famodiff.tex	Thu Jun 03 09:47:18 2010 -0700
@@ -207,7 +207,7 @@
 Since PL homeomorphisms are bi-Lipschitz, we have established this last remaining case of claim 4 of the lemma as well.
 \end{proof}
 
-\begin{lemma}
+\begin{lemma} \label{extension_lemma_c}
 Let $\cX_*$ be any of $C_*(\Maps(X \to T))$ or singular chains on the subspace of $\Maps(X\to T)$ consisting of immersions, diffeomorphisms, bi-Lipschitz homeomorphisms or PL homeomorphisms.
 Let $G_* \subset \cX_*$ denote the chains adapted to an open cover $\cU$
 of $X$.
diff -r d163ad9543a5 -r bc22926d4fb0 text/comm_alg.tex
--- a/text/comm_alg.tex	Wed Jun 02 17:45:13 2010 -0700
+++ b/text/comm_alg.tex	Thu Jun 03 09:47:18 2010 -0700
@@ -3,8 +3,7 @@
 \section{Commutative algebras as $n$-categories}
 \label{sec:comm_alg}
 
-\nn{this should probably not be a section by itself.  i'm just trying to write down the outline 
-while it's still fresh in my mind.}
+\nn{should consider leaving this out; for now, make it an appendix.}
 
 \nn{also, this section needs a little updating to be compatible with the rest of the paper.}
 
diff -r d163ad9543a5 -r bc22926d4fb0 text/ncat.tex
--- a/text/ncat.tex	Wed Jun 02 17:45:13 2010 -0700
+++ b/text/ncat.tex	Thu Jun 03 09:47:18 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 ($k\le n$),