786 \end{thm} |
786 \end{thm} |
787 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
787 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
788 (see \cite[\S7.1]{1009.5025}). |
788 (see \cite[\S7.1]{1009.5025}). |
789 |
789 |
790 \begin{proof} (Sketch.) |
790 \begin{proof} (Sketch.) |
791 |
791 The proof is similar to that of the second part of Theorem \ref{thm:gluing}. |
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792 There is a natural map from the 0-simplices of $\clh{\bc_*(Y;\cC)}(W)$ to $\bc_*(Y\times W; \cC)$, |
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793 given by reinterpreting a decomposition of $W$ labeled by $(n{-}k)$-morphisms of $\bc_*(Y; \cC)$ as a blob |
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794 diagram on $W\times Y$. |
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795 This map can be extended to all of $\clh{\bc_*(Y;\cC)}(W)$ by sending higher simplices to zero. |
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796 |
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797 To construct the homotopy inverse of the above map one first shows that |
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798 $\bc_*(Y\times W; \cC)$ is homotopy equivalent to the subcomplex generated by blob diagrams which |
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799 are small with respect any fixed open cover of $Y\times W$. |
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800 For a sufficiently fine open cover the generators of this ``small" blob complex are in the image of the map |
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801 of the previous paragraph, and furthermore the preimage in $\clh{\bc_*(Y;\cC)}(W)$ of such small diagrams |
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802 lie in contractible subcomplexes. |
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803 A standard acyclic models argument now constructs the homotopy inverse. |
792 \end{proof} |
804 \end{proof} |
793 |
805 |
794 %\nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.} |
806 %\nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.} |
795 |
807 |
796 \section{Higher Deligne conjecture} |
808 \section{Deligne conjecture for $n$-categories} |
797 \label{sec:applications} |
809 \label{sec:applications} |
798 |
810 |
799 \begin{thm}[Higher dimensional Deligne conjecture] |
811 \begin{thm}[Higher dimensional Deligne conjecture] |
800 \label{thm:deligne} |
812 \label{thm:deligne} |
801 The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. |
813 The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. |
816 This follows from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and associativity. |
828 This follows from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and associativity. |
817 \end{proof} |
829 \end{proof} |
818 |
830 |
819 The little disks operad $LD$ is homotopy equivalent to |
831 The little disks operad $LD$ is homotopy equivalent to |
820 \nn{suboperad of} |
832 \nn{suboperad of} |
821 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 cohains $Hoch^*(C, C)$. |
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 $Hoch^*(C, C)$. |
822 The usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923}) gives a map |
834 The usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923}) gives a map |
823 \[ |
835 \[ |
824 C_*(LD_k)\tensor \overbrace{Hoch^*(C, C)\tensor\cdots\tensor Hoch^*(C, C)}^{\text{$k$ copies}} |
836 C_*(LD_k)\tensor \overbrace{Hoch^*(C, C)\tensor\cdots\tensor Hoch^*(C, C)}^{\text{$k$ copies}} |
825 \to Hoch^*(C, C), |
837 \to Hoch^*(C, C), |
826 \] |
838 \] |