--- a/blob1.tex Tue Jun 01 11:34:03 2010 -0700
+++ b/blob1.tex Tue Jun 01 17:26:28 2010 -0700
@@ -25,24 +25,6 @@
\textbf{Draft version, read with caution.}
-\medskip
-
-\nn{need to remove or at least water down this warning for the arxiv version}
-
-\noindent
-{\bf Warning:} This draft is draftier than you might expect.
-More specifically,
-\begin{itemize}
-\item Some sections are missing.
-\item Many sections are incomplete.
-In most cases the incompleteness is noted, but occasionally it isn't.
-\item Some sections were written nearly two years ago, and are now outdated.
-\item There are not yet enough citations to similar work of other people.
-\end{itemize}
-Despite all this, there's probably enough decipherable material
-here to interest the motivated reader.
-On the other hand, if you are only going to read this paper once,
-{\bf then don't read this version,} as a more complete version will be available in a couple of months.
\nn{maybe to do: add appendix on various versions of acyclic models}
--- a/text/a_inf_blob.tex Tue Jun 01 11:34:03 2010 -0700
+++ b/text/a_inf_blob.tex Tue Jun 01 17:26:28 2010 -0700
@@ -208,7 +208,7 @@
This is clear, since the acyclic subcomplexes $D(a)$ above used in the definition of
$\phi$ and $\phi'$ do not depend on the choice of cover.
-\nn{need to say (and justify) that we now have a map $\phi$ indep of choice of cover}
+%\nn{need to say (and justify) that we now have a map $\phi$ indep of choice of cover}
We now show that $\phi\circ\psi$ and $\psi\circ\phi$ are homotopic to the identity.
@@ -217,10 +217,10 @@
according to some decomposition $K$ of $Y$.
$\psi$ glues those pieces back together, yielding the same $a$ we started with.
-$\phi\circ\psi$ is the identity up to homotopy by another MoAM argument...
+$\phi\circ\psi$ is the identity up to homotopy by another MoAM argument....
+
This concludes the proof of Theorem \ref{product_thm}.
-\nn{at least I think it does; it's pretty rough at this point.}
\end{proof}
\nn{need to say something about dim $< n$ above}
@@ -237,21 +237,31 @@
\medskip
-\nn{To do: remark on the case of a nontrivial fiber bundle.
-I can think of two approaches.
-In the first (slick but maybe a little too tautological), we generalize the
-notion of an $n$-category to an $n$-category {\it over a space $B$}.
-(Should be able to find precedent for this in a paper of PT.
-This idea came up in a conversation with him, so maybe should site him.)
-In this generalization, we replace the categories of balls with the categories
-of balls equipped with maps to $B$.
-A fiber bundle $F\to E\to B$ gives an example of such an $n$-category:
-assign to $p:D\to B$ the blob complex $\bc_*(p^*(E))$.
-We can do the colimit thing over $B$ with coefficients in a n-cat-over-B.
-The proof below works essentially unchanged in this case to show that the colimit is the blob complex of the total space $E$.
-}
+Theorem \ref{product_thm} extends to the case of general fiber bundles
+\[
+ F \to E \to Y .
+\]
+We outline two approaches.
-\nn{The second approach: Choose a decomposition $B = \cup X_i$
+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?}
+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 $\bc_*(p^*(E))$.
+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
+to show that
+\[
+ \bc_*(E) \simeq \cF_E(Y) .
+\]
+
+
+
+
+\nn{The second approach: Choose a decomposition $Y = \cup X_i$
such that the restriction of $E$ to $X_i$ is a product $F\times X_i$.
Choose the product structure as well.
To each codim-1 face $D_i\cap D_j$ we have a bimodule ($S^0$-module).
@@ -259,7 +269,7 @@
Decorate the decomposition with these modules and do the colimit.
}
-\nn{There is a version of this last construction for arbitrary maps $E \to B$
+\nn{There is a version of this last construction for arbitrary maps $E \to Y$
(not necessarily a fibration).}