
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




