165 \nn{add product formula?  $n$-dimensional fat graph operad stuff?}

   165 

   166 

   167 \begin{property}[Relation to mapping spaces]

   168 There is a version of the blob complex for $C$ an $A_\infty$ $n$-category

   169 instead of a garden variety $n$-category.

   170 

   171 Let $\pi^\infty_{\le n}(W)$ denote the $A_\infty$ $n$-category based on maps 

   172 $B^n \to W$.

   173 (The case $n=1$ is the usual $A_\infty$ category of paths in $W$.)

   174 Then $\bc_*(M, \pi^\infty_{\le n}(W))$ is 

   175 homotopy equivalent to $C_*(\{\text{maps}\; M \to W\})$.

   176 \end{property}

   177 

   178 

   179 

   180 

   181 \begin{property}[Product formula]

   182 Let $M^n = Y^{n-k}\times W^k$ and let $C$ be an $n$-category.

   183 Let $A_*(Y)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology.

   184 Then

   185 \[

   186 	\bc_*(Y^{n-k}\times W^k, C) \simeq \bc_*(W, A_*(Y)) .

   187 \]

   188 \nn{say something about general fiber bundles?}

   189 \end{property}

   190 

   191 

   192 

   193 

   194 \begin{property}[Higher dimensional Deligne conjecture]

   195 The singular chains of the $n$-dimensional fat graph operad act on blob cochains.

   196 

   197 The $n$-dimensional fat graph operad can be thought of as a sequence of general surgeries

   198 of $n$-manifolds

   199 $R_i \cup A_i \leadsto R_i \cup B_i$ together with mapping cylinders of diffeomorphisms

   200 $f_i: R_i\cup B_i \to R_{i+1}\cup A_{i+1}$.

   201 (Note that the suboperad where $A_i$, $B_i$ and $R_i\cup A_i$ are all diffeomorphic to 

   202 the $n$-ball is equivalent to the little $n{+}1$-disks operad.)

   203 

   204 If $A$ and $B$ are $n$-manifolds sharing the same boundary, define

   205 the blob cochains $\bc^*(A, B)$ (analogous to Hochschild cohomology) to be

   206 $A_\infty$ maps from $\bc_*(A)$ to $\bc_*(B)$, where we think of both

   207 (collections of) complexes as modules over the $A_\infty$ category associated to $\bd A = \bd B$.

   208 The ``holes" in the above 

   209 $n$-dimensional fat graph operad are labeled by $\bc^*(A_i, B_i)$.

   210 \end{property}

   211 

   212 

   213 

   214 

   215 

   216 

   217