# HG changeset patch
# User kevin@6e1638ff-ae45-0410-89bd-df963105f760
# Date 1251254124 0
# Node ID b62214646c4f858712267dcf058582e4af0c3bdd
# Parent  3f180943709f4ab8125358a0065ffd9add95646e
preparing for semi-public version soon

diff -r 3f180943709f -r b62214646c4f blob1.tex
--- a/blob1.tex	Wed Aug 26 01:21:59 2009 +0000
+++ b/blob1.tex	Wed Aug 26 02:35:24 2009 +0000
@@ -22,7 +22,23 @@
 %\versioninfo
 [26 August 2009]
 
+\medskip
 
+\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 some cases the incompleteness is noted, in some cases not.
+\item Some sections have been rewritten, but the older, obsolete version of
+the section has not been deleted yet.
+\item Some sections were written nearly two years ago, and are now outdated.
+\item Some sections have not been proof-read.
+\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.
 
 \noop{
 
diff -r 3f180943709f -r b62214646c4f text/a_inf_blob.tex
--- a/text/a_inf_blob.tex	Wed Aug 26 01:21:59 2009 +0000
+++ b/text/a_inf_blob.tex	Wed Aug 26 02:35:24 2009 +0000
@@ -98,6 +98,7 @@
 We want to show that this cycle bounds a chain of filtration degree 2 stuff.
 Choose a decomposition $M$ which has common refinements with each of 
 $K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$.
+\nn{need to also require that $KLM$ antirefines to $KM$, etc.}
 Then we have a filtration degree 2 chain, as shown in Figure yyyy, which does the trick.
 For example, ....
 
diff -r 3f180943709f -r b62214646c4f text/comm_alg.tex
--- a/text/comm_alg.tex	Wed Aug 26 01:21:59 2009 +0000
+++ b/text/comm_alg.tex	Wed Aug 26 02:35:24 2009 +0000
@@ -5,6 +5,12 @@
 \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{I strongly suspect that [blob complex
+for $M^n$ based on comm alg $C$ thought of as an $n$-category]
+is homotopy equivalent to [higher Hochschild complex for $M^n$ with coefficients in $C$].
+(Thomas Tradler's idea.)
+Should prove (or at least conjecture) that here.}
+
 If $C$ is a commutative algebra it
 can (and will) also be thought of as an $n$-category whose $j$-morphisms are trivial for
 $j<n$ and whose $n$-morphisms are $C$. 
diff -r 3f180943709f -r b62214646c4f text/definitions.tex
--- a/text/definitions.tex	Wed Aug 26 01:21:59 2009 +0000
+++ b/text/definitions.tex	Wed Aug 26 02:35:24 2009 +0000
@@ -3,6 +3,9 @@
 \section{Definitions}
 \label{sec:definitions}
 
+\nn{this section is a bit out of date; needs to be updated 
+to fit with $n$-category definition given later}
+
 \subsection{Systems of fields}
 \label{sec:fields}
 
@@ -19,7 +22,7 @@
 is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$
 together with some additional data and satisfying some additional conditions, all specified below.
 
-\nn{refer somewhere to my TQFT notes \cite{kw:tqft}, and possibly also to paper with Chris}
+\nn{refer somewhere to my TQFT notes \cite{kw:tqft}}
 
 Before finishing the definition of fields, we give two motivating examples
 (actually, families of examples) of systems of fields.
diff -r 3f180943709f -r b62214646c4f text/intro.tex
--- a/text/intro.tex	Wed Aug 26 01:21:59 2009 +0000
+++ b/text/intro.tex	Wed Aug 26 02:35:24 2009 +0000
@@ -162,10 +162,63 @@
 \end{itemize}
 \end{property}
 
-\nn{add product formula?  $n$-dimensional fat graph operad stuff?}
+
+
+\begin{property}[Relation to mapping spaces]
+There is a version of the blob complex for $C$ an $A_\infty$ $n$-category
+instead of a garden variety $n$-category.
+
+Let $\pi^\infty_{\le n}(W)$ denote the $A_\infty$ $n$-category based on maps 
+$B^n \to W$.
+(The case $n=1$ is the usual $A_\infty$ category of paths in $W$.)
+Then $\bc_*(M, \pi^\infty_{\le n}(W))$ is 
+homotopy equivalent to $C_*(\{\text{maps}\; M \to W\})$.
+\end{property}
+
+
+
+
+\begin{property}[Product formula]
+Let $M^n = Y^{n-k}\times W^k$ and let $C$ be an $n$-category.
+Let $A_*(Y)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology.
+Then
+\[
+	\bc_*(Y^{n-k}\times W^k, C) \simeq \bc_*(W, A_*(Y)) .
+\]
+\nn{say something about general fiber bundles?}
+\end{property}
+
+
+
+
+\begin{property}[Higher dimensional Deligne conjecture]
+The singular chains of the $n$-dimensional fat graph operad act on blob cochains.
+
+The $n$-dimensional fat graph operad can be thought of as a sequence of general surgeries
+of $n$-manifolds
+$R_i \cup A_i \leadsto R_i \cup B_i$ together with mapping cylinders of diffeomorphisms
+$f_i: R_i\cup B_i \to R_{i+1}\cup A_{i+1}$.
+(Note that the suboperad where $A_i$, $B_i$ and $R_i\cup A_i$ are all diffeomorphic to 
+the $n$-ball is equivalent to the little $n{+}1$-disks operad.)
+
+If $A$ and $B$ are $n$-manifolds sharing the same boundary, define
+the blob cochains $\bc^*(A, B)$ (analogous to Hochschild cohomology) to be
+$A_\infty$ maps from $\bc_*(A)$ to $\bc_*(B)$, where we think of both
+(collections of) complexes as modules over the $A_\infty$ category associated to $\bd A = \bd B$.
+The ``holes" in the above 
+$n$-dimensional fat graph operad are labeled by $\bc^*(A_i, B_i)$.
+\end{property}
+
+
+
+
+
+
+
 
 Properties \ref{property:functoriality}, \ref{property:gluing-map} and \ref{property:skein-modules} will be immediate from the definition given in
 \S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. \todo{Make sure this gets done.}
 Properties \ref{property:disjoint-union} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}.
 Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation},
 and Property \ref{property:gluing} in \S \ref{sec:gluing}.
+\nn{need to say where the remaining properties are proved.}
\ No newline at end of file
diff -r 3f180943709f -r b62214646c4f text/ncat.tex
--- a/text/ncat.tex	Wed Aug 26 01:21:59 2009 +0000
+++ b/text/ncat.tex	Wed Aug 26 02:35:24 2009 +0000
@@ -457,7 +457,8 @@
 $\psi_\cC(x)\to \cC(W)$, these maps are compatible with the refinement maps
 above, and $\cC(W)$ is universal with respect to these properties.
 In the $A_\infty$ case, it means 
-\nn{.... need to check if there is a def in the literature before writing this down}
+\nn{.... need to check if there is a def in the literature before writing this down;
+homotopy colimit I think}
 
 More concretely, in the plain case enriched over vector spaces, and with $\dim(W) = n$, we can take
 \[
@@ -469,6 +470,7 @@
 
 In the $A_\infty$ case enriched over chain complexes, the concrete description of the colimit
 is as follows.
+\nn{should probably rewrite this to be compatible with some standard reference}
 Define an $m$-sequence to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions.
 Such sequences (for all $m$) form a simplicial set.
 Let
@@ -815,6 +817,12 @@
 
 
 
+\subsection{The $n{+}1$-category of sphere modules}
+
+Outline:
+\begin{itemize}
+\item 
+\end{itemize}
 
 
 
@@ -838,7 +846,9 @@
 \item say something about unoriented vs oriented vs spin vs pin for $n=1$ (and $n=2$?)
 \item spell out what difference (if any) Top vs PL vs Smooth makes
 \item explain relation between old-fashioned blob homology and new-fangled blob homology
-\item define $n{+}1$-cat of $n$-cats; discuss Morita equivalence
+(follows as special case of product formula (product with a point).
+\item define $n{+}1$-cat of $n$-cats (a.k.a.\ $n{+}1$-category of generalized bimodules
+a.k.a.\ $n{+}1$-category of sphere modules); discuss Morita equivalence
 \end{itemize}