# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1289955355 28800
# Node ID da5077cae33ce2f701a5c3cac1e11146fa235299
# Parent  a1fa4428ddbc54c2c31a3e117c4ad704e95d04cc# Parent  e0093da0d39f7a1abc3eed209324ce1e047aa1a2
Automated merge with https://tqft.net/hg/blob

diff -r e0093da0d39f -r da5077cae33c pnas/pnas.tex
--- a/pnas/pnas.tex	Tue Nov 16 16:55:49 2010 -0800
+++ b/pnas/pnas.tex	Tue Nov 16 16:55:55 2010 -0800
@@ -266,7 +266,7 @@
 Thus we can have the simplicity of strict associativity in exchange for more morphisms.
 We wish to imitate this strategy in higher categories.
 Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with
-a product of $k$ intervals \nn{cf xxxx} but rather with any $k$-ball, that is, any $k$-manifold which is homeomorphic
+a product of $k$ intervals (c.f. \cite{0909.2212}) but rather with any $k$-ball, that is, any $k$-manifold which is homeomorphic
 to the standard $k$-ball $B^k$.
 \nn{maybe add that in addition we want functoriality}
 
@@ -292,7 +292,7 @@
 As such, we don't subdivide the boundary of a morphism
 into domain and range --- the duality operations can convert between domain and range.
 
-Later \todo{} we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$ from arbitrary manifolds to sets. We need the restriction of these functors to $k$-spheres, for $k<n$, for the next axiom.
+Later \nn{make sure this actually happens, or reorganise} we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$ from arbitrary manifolds to sets. We need the restriction of these functors to $k$-spheres, for $k<n$, for the next axiom.
 
 \begin{axiom}[Boundaries]\label{nca-boundary}
 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$.
@@ -534,7 +534,7 @@
 
 \subsubsection{Homotopy colimits}
 \nn{Motivation: How can we extend an $n$-category from balls to arbitrary manifolds?}
-\todo{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls?}
+\nn{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls?}
 \nn{Explain codimension colimits here too}
 
 We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$
@@ -545,7 +545,7 @@
 $$\bdy (\bar{x},a) = (\bar{x}, \bdy a) + (-1)^{\deg a} \left( (d_0 \bar{x}, g(a)) + \sum_{i=1}^m (-1)^i (d_i \bar{x}, a) \right)$$
 where $g$ is the gluing map from $x_0$ to $x_1$, and $d_i \bar{x}$ denotes the $i$-th face of the simplex $\bar{x}$.
 
-Alternatively, we can take advantage of the product structure on $\cell(W)$ to realize the homotopy colimit via the cone-product polyhedra in $\cell(W)$. A cone-product polyhedra is obtained from a point by successively taking the cone or taking the product with another cone-product polyhedron. Just as simplices correspond to linear directed graphs, cone-product polyheda correspond to directed trees: taking cone adds a new root before the existing root, and taking product identifies the roots of several trees. The `local homotopy colimit' is then defined according to the same formula as above, but with $x$ a cone-product polyhedron in $\cell(W)$.
+Alternatively, we can take advantage of the product structure on $\cell(W)$ to realize the homotopy colimit via the cone-product polyhedra in $\cell(W)$. A cone-product polyhedra is obtained from a point by successively taking the cone or taking the product with another cone-product polyhedron. Just as simplices correspond to linear directed graphs, cone-product polyheda correspond to directed trees: taking cone adds a new root before the existing root, and taking product identifies the roots of several trees. The `local homotopy colimit' is then defined according to the same formula as above, but with $\bar{x}$ a cone-product polyhedron in $\cell(W)$. The differential acts on $(\bar{x},a)$ both on $a$ and on $\bar{x}$, applying the appropriate gluing map to $a$ when required.
 A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization.
 
 %When $\cC$ is a topological $n$-category,
@@ -667,10 +667,7 @@
 \xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.47){\iso}_(0.47){\text{qi}} & \HC_*(\cC).}
 \end{equation*}
 \end{thm}
-
-Theorem \ref{thm:skein-modules} is immediate from the definition, and
-Theorem \ref{thm:hochschild} is established by extending the statement to bimodules as well as categories, then verifying that the universal properties of Hochschild homology also hold for $\bc_*(S^1; -)$.
-
+This theorem is established by extending the statement to bimodules as well as categories, then verifying that the universal properties of Hochschild homology also hold for $\bc_*(S^1; -)$.
 
 \begin{thm}[Mapping spaces]
 \label{thm:map-recon}
@@ -682,9 +679,7 @@
 \end{thm}
 
 This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. 
-Note that there is no restriction on the connectivity of $T$ as there is for the corresponding result in topological chiral homology \cite[Theorem 3.8.6]{0911.0018}.
-\todo{sketch proof}
-
+Note that there is no restriction on the connectivity of $T$ as there is for the corresponding result in topological chiral homology \cite[Theorem 3.8.6]{0911.0018}. The result was proved in \cite[\S 7.3]{1009.5025}.
 
 \subsection{Structure of the blob complex}
 \label{sec:structure}
@@ -873,10 +868,7 @@
 We have that $SC^1_{\overline{M}, \overline{N}}$ is homotopy equivalent to the little
 disks operad and $\hom(\bc_*(M_i), \bc_*(N_i))$ is homotopy equivalent to Hochschild cochains.
 This special case is just the usual Deligne conjecture
-(see \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923} 
-\nn{should check that this is the optimal list of references; what about Gerstenhaber-Voronov?;
-if we revise this list, should propagate change back to main paper}
-).
+(see \cite{hep-th/9403055, MR1328534, MR1805894, MR1805923, MR2064592}).
 
 The general case when $n=1$ goes beyond the original Deligne conjecture, as the $M_i$'s and $N_i$'s
 could be disjoint unions of 1-balls and circles, and the surgery cylinders could be high genus surfaces.
diff -r e0093da0d39f -r da5077cae33c text/deligne.tex
--- a/text/deligne.tex	Tue Nov 16 16:55:49 2010 -0800
+++ b/text/deligne.tex	Tue Nov 16 16:55:55 2010 -0800
@@ -12,7 +12,7 @@
 %Different versions of the geometric counterpart of Deligne's conjecture have been proven by Tamarkin [``Formality of chain operad of small squares'', preprint, http://arXiv.org/abs/math.QA/9809164], the reviewer [in Confˇrence Moshˇ Flato 1999, Vol. II (Dijon), 307--331, Kluwer Acad. Publ., Dordrecht, 2000; MR1805923 (2002d:55009)], and J. E. McClure and J. H. Smith [``A solution of Deligne's conjecture'', preprint, http://arXiv.org/abs/math.QA/9910126] (see also a later simplified version [J. E. McClure and J. H. Smith, ``Multivariable cochain operations and little $n$-cubes'', preprint, http://arXiv.org/abs/math.QA/0106024]). The paper under review gives another proof of Deligne's conjecture, which, as the authors indicate, may be generalized to a proof of a higher-dimensional generalization of Deligne's conjecture, suggested in [M. Kontsevich, Lett. Math. Phys. 48 (1999), no. 1, 35--72; MR1718044 (2000j:53119)]. 
 
 
-The usual Deligne conjecture (proved variously in \cite{MR1805894, MR2064592, hep-th/9403055, MR1805923}) gives a map
+The usual Deligne conjecture (proved variously in \cite{MR1805894, MR1328534, MR2064592, hep-th/9403055, MR1805923}) gives a map
 \[
 	C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}}
 			\to  Hoch^*(C, C) .