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216 \nn{Triangulated categories are important; often calculations are via exact sequences, |
216 \nn{Triangulated categories are important; often calculations are via exact sequences, |
217 and the standard TQFT constructions are quotients, which destroy exactness.} |
217 and the standard TQFT constructions are quotients, which destroy exactness.} |
218 |
218 |
219 \nn{In many places we omit details; they can be found in MW. |
219 \nn{In many places we omit details; they can be found in MW. |
220 (Blanket statement in order to avoid too many citations to MW.)} |
220 (Blanket statement in order to avoid too many citations to MW.)} |
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221 |
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222 \nn{perhaps say something explicit about the relationship of this paper to big blob paper. |
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223 like: in this paper we try to give a clear view of the big picture without getting bogged down in details} |
221 |
224 |
222 \section{Definitions} |
225 \section{Definitions} |
223 \subsection{$n$-categories} \mbox{} |
226 \subsection{$n$-categories} \mbox{} |
224 |
227 |
225 \nn{rough draft of n-cat stuff...} |
228 \nn{rough draft of n-cat stuff...} |
829 \end{proof} |
832 \end{proof} |
830 |
833 |
831 The little disks operad $LD$ is homotopy equivalent to |
834 The little disks operad $LD$ is homotopy equivalent to |
832 \nn{suboperad of} |
835 \nn{suboperad of} |
833 the $n=1$ case of the $n$-SC operad. The blob complex $\bc_*(I, \cC)$ is a bimodule over itself, and the $A_\infty$-bimodule intertwiners are homotopy equivalent to the Hochschild cochains $Hoch^*(C, C)$. |
836 the $n=1$ case of the $n$-SC operad. The blob complex $\bc_*(I, \cC)$ is a bimodule over itself, and the $A_\infty$-bimodule intertwiners are homotopy equivalent to the Hochschild cochains $Hoch^*(C, C)$. |
834 The usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923}) gives a map |
837 The usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923}) |
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838 \nn{should check that this is the optimal list of references; what about Gerstenhaber-Voronov?; |
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839 if we revise this list, should propagate change back to main paper} |
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840 gives a map |
835 \[ |
841 \[ |
836 C_*(LD_k)\tensor \overbrace{Hoch^*(C, C)\tensor\cdots\tensor Hoch^*(C, C)}^{\text{$k$ copies}} |
842 C_*(LD_k)\tensor \overbrace{Hoch^*(C, C)\tensor\cdots\tensor Hoch^*(C, C)}^{\text{$k$ copies}} |
837 \to Hoch^*(C, C), |
843 \to Hoch^*(C, C), |
838 \] |
844 \] |
839 which we now see to be a specialization of Theorem \ref{thm:deligne}. |
845 which we now see to be a specialization of Theorem \ref{thm:deligne}. |