237 We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second. |
237 We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second. |
238 More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.} |
238 More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.} |
239 |
239 |
240 \nn{say something about defining plain and infty cases simultaneously} |
240 \nn{say something about defining plain and infty cases simultaneously} |
241 |
241 |
242 There are five basic ingredients of an $n$-category definition: |
242 There are five basic ingredients |
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243 (not two, or four, or seven, but {\bf five} basic ingredients, |
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244 which he shall wield all wretched sinners and that includes on you, sir, there in the front row! |
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245 (cf.\ Monty Python, Life of Brian, http://www.youtube.com/watch?v=fIRb8TigJ28)) |
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246 of an $n$-category definition: |
243 $k$-morphisms (for $0\le k \le n$), domain and range, composition, |
247 $k$-morphisms (for $0\le k \le n$), domain and range, composition, |
244 identity morphisms, and special behavior in dimension $n$ (e.g. enrichment |
248 identity morphisms, and special behavior in dimension $n$ (e.g. enrichment |
245 in some auxiliary category, or strict associativity instead of weak associativity). |
249 in some auxiliary category, or strict associativity instead of weak associativity). |
246 We will treat each of these in turn. |
250 We will treat each of these in turn. |
247 |
251 |
538 %the flexibility available in the construction of a homotopy colimit allows |
542 %the flexibility available in the construction of a homotopy colimit allows |
539 %us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. |
543 %us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. |
540 %\todo{either need to explain why this is the same, or significantly rewrite this section} |
544 %\todo{either need to explain why this is the same, or significantly rewrite this section} |
541 When $\cC$ is the topological $n$-category based on string diagrams for a traditional |
545 When $\cC$ is the topological $n$-category based on string diagrams for a traditional |
542 $n$-category $C$, |
546 $n$-category $C$, |
543 one can show \nn{cite us} that the above two constructions of the homotopy colimit |
547 one can show \cite{1009.5025} that the above two constructions of the homotopy colimit |
544 are equivalent to the more concrete construction which we describe next, and which we denote $\bc_*(W; \cC)$. |
548 are equivalent to the more concrete construction which we describe next, and which we denote $\bc_*(W; \cC)$. |
545 Roughly speaking, the generators of $\bc_k(W; \cC)$ are string diagrams on $W$ together with |
549 Roughly speaking, the generators of $\bc_k(W; \cC)$ are string diagrams on $W$ together with |
546 a configuration of $k$ balls (or ``blobs") in $W$ whose interiors are pairwise disjoint or nested. |
550 a configuration of $k$ balls (or ``blobs") in $W$ whose interiors are pairwise disjoint or nested. |
547 The restriction of the string diagram to innermost blobs is required to be ``null" in the sense that |
551 The restriction of the string diagram to innermost blobs is required to be ``null" in the sense that |
548 it evaluates to a zero $n$-morphism of $C$. |
552 it evaluates to a zero $n$-morphism of $C$. |
716 \begin{thm} |
720 \begin{thm} |
717 \label{thm:blobs-ainfty} |
721 \label{thm:blobs-ainfty} |
718 Let $\cC$ be a topological $n$-category. |
722 Let $\cC$ be a topological $n$-category. |
719 Let $Y$ be an $n{-}k$-manifold. |
723 Let $Y$ be an $n{-}k$-manifold. |
720 There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, |
724 There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, |
721 to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set |
725 to be the set $$\bc_*(Y;\cC)(D) = \cl{\cC}(Y \times D)$$ and on $k$-balls $D$ to be the set |
722 $$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ |
726 $$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ |
723 (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) |
727 (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) |
724 These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in |
728 These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in |
725 Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}. |
729 Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}. |
726 \end{thm} |
730 \end{thm} |
727 \begin{rem} |
731 \begin{rem} |
728 When $Y$ is a point this gives $A_\infty$ $n$-category from a topological $n$-category, which can be thought of as a free resolution. |
732 When $Y$ is a point this produces an $A_\infty$ $n$-category from a topological $n$-category, |
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733 which can be thought of as a free resolution. |
729 \end{rem} |
734 \end{rem} |
730 This result is described in more detail as Example 6.2.8 of \cite{1009.5025} |
735 This result is described in more detail as Example 6.2.8 of \cite{1009.5025}. |
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736 |
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737 Fix a topological $n$-category $\cC$, which we'll now omit from notation. |
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738 Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. |
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739 |
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740 \begin{thm}[Gluing formula] |
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741 \label{thm:gluing} |
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742 \mbox{}% <-- gets the indenting right |
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743 \begin{itemize} |
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744 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an |
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745 $A_\infty$ module for $\bc_*(Y)$. |
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746 |
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747 \item The blob complex of a glued manifold $X\bigcup_Y \selfarrow$ is the $A_\infty$ self-tensor product of |
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748 $\bc_*(X)$ as an $\bc_*(Y)$-bimodule: |
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749 \begin{equation*} |
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750 \bc_*(X\bigcup_Y \selfarrow) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
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751 \end{equation*} |
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752 \end{itemize} |
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753 \end{thm} |
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754 |
731 |
755 |
732 We next describe the blob complex for product manifolds, in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as above. |
756 We next describe the blob complex for product manifolds, in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as above. |
733 |
757 |
734 \begin{thm}[Product formula] |
758 \begin{thm}[Product formula] |
735 \label{thm:product} |
759 \label{thm:product} |
741 \bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W). |
765 \bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W). |
742 \] |
766 \] |
743 \end{thm} |
767 \end{thm} |
744 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
768 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
745 (see \cite[\S7.1]{1009.5025}). |
769 (see \cite[\S7.1]{1009.5025}). |
746 |
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747 Fix a topological $n$-category $\cC$, which we'll now omit from notation. |
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748 Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. |
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749 |
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750 \begin{thm}[Gluing formula] |
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751 \label{thm:gluing} |
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752 \mbox{}% <-- gets the indenting right |
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753 \begin{itemize} |
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754 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an |
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755 $A_\infty$ module for $\bc_*(Y)$. |
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756 |
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757 \item The blob complex of a glued manifold $X\bigcup_Y \selfarrow$ is the $A_\infty$ self-tensor product of |
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758 $\bc_*(X)$ as an $\bc_*(Y)$-bimodule: |
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759 \begin{equation*} |
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760 \bc_*(X\bigcup_Y \selfarrow) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
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761 \end{equation*} |
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762 \end{itemize} |
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763 \end{thm} |
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764 |
770 |
765 \nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.} |
771 \nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.} |
766 |
772 |
767 \section{Applications} |
773 \section{Applications} |
768 \label{sec:applications} |
774 \label{sec:applications} |
786 The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. |
792 The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. |
787 Since the little $n{+}1$-balls operad is a suboperad of the $n$-SC operad, |
793 Since the little $n{+}1$-balls operad is a suboperad of the $n$-SC operad, |
788 this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball. |
794 this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball. |
789 \end{thm} |
795 \end{thm} |
790 |
796 |
791 An $n$-dimensional surgery cylinder is a sequence of mapping cylinders and surgeries (Figure \ref{delfig2}), modulo changing the order of distant surgeries, and conjugating a submanifold not modified in a surgery by a homeomorphism. Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another. |
797 An $n$-dimensional surgery cylinder is a sequence of mapping cylinders and surgeries (Figure \ref{delfig2}), |
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798 modulo changing the order of distant surgeries, and conjugating a submanifold not modified in a surgery by a homeomorphism. |
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799 Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another. |
792 |
800 |
793 By the `blob cochains' of a manifold $X$, we mean the $A_\infty$ maps of $\bc_*(X)$ as a $\bc_*(\bdy X)$ $A_\infty$-module. |
801 By the `blob cochains' of a manifold $X$, we mean the $A_\infty$ maps of $\bc_*(X)$ as a $\bc_*(\bdy X)$ $A_\infty$-module. |
794 |
802 |
795 \begin{proof} |
803 \begin{proof} |
796 We have already defined the action of mapping cylinders, in Theorem \ref{thm:evaluation}, and the action of surgeries is just composition of maps of $A_\infty$-modules. We only need to check that the relations of the $n$-SC operad are satisfied. This follows immediately from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and associativity. |
804 We have already defined the action of mapping cylinders, in Theorem \ref{thm:evaluation}, |
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805 and the action of surgeries is just composition of maps of $A_\infty$-modules. |
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806 We only need to check that the relations of the $n$-SC operad are satisfied. |
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807 This follows from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and associativity. |
797 \end{proof} |
808 \end{proof} |
798 |
809 |
799 The little disks operad $LD$ is homotopy equivalent to 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 $Hoch^*(C, C)$. The usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923}) gives a map |
810 The little disks operad $LD$ is homotopy equivalent to |
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811 \nn{suboperad of} |
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812 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 $Hoch^*(C, C)$. |
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813 The usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923}) gives a map |
800 \[ |
814 \[ |
801 C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}} |
815 C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}} |
802 \to Hoch^*(C, C), |
816 \to Hoch^*(C, C), |
803 \] |
817 \] |
804 which we now see to be a specialization of Theorem \ref{thm:deligne}. |
818 which we now see to be a specialization of Theorem \ref{thm:deligne}. |
819 %% \appendix Appendix text... |
833 %% \appendix Appendix text... |
820 %% or, for appendix with title, use square brackets: |
834 %% or, for appendix with title, use square brackets: |
821 %% \appendix[Appendix Title] |
835 %% \appendix[Appendix Title] |
822 |
836 |
823 \begin{acknowledgments} |
837 \begin{acknowledgments} |
824 \nn{say something here} |
838 It is a pleasure to acknowledge helpful conversations with |
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839 Kevin Costello, |
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840 Mike Freedman, |
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841 Justin Roberts, |
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842 and |
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843 Peter Teichner. |
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844 \nn{not full list from big paper, but only most significant names} |
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845 We also thank the Aspen Center for Physics for providing a pleasant and productive |
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846 environment during the last stages of this project. |
825 \end{acknowledgments} |
847 \end{acknowledgments} |
826 |
848 |
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