237 More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.} |
237 More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.} |
238 |
238 |
239 \nn{say something about defining plain and infty cases simultaneously} |
239 \nn{say something about defining plain and infty cases simultaneously} |
240 |
240 |
241 There are five basic ingredients |
241 There are five basic ingredients |
242 (not two, or four, or seven, but {\bf five} basic ingredients, |
242 \cite{life-of-brian} of an $n$-category definition: |
243 which he shall wield all wretched sinners and that includes on you, sir, there in the front row! |
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244 (cf.\ Monty Python, Life of Brian, http://www.youtube.com/watch?v=fIRb8TigJ28)) |
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245 of an $n$-category definition: |
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246 $k$-morphisms (for $0\le k \le n$), domain and range, composition, |
243 $k$-morphisms (for $0\le k \le n$), domain and range, composition, |
247 identity morphisms, and special behavior in dimension $n$ (e.g. enrichment |
244 identity morphisms, and special behavior in dimension $n$ (e.g. enrichment |
248 in some auxiliary category, or strict associativity instead of weak associativity). |
245 in some auxiliary category, or strict associativity instead of weak associativity). |
249 We will treat each of these in turn. |
246 We will treat each of these in turn. |
250 |
247 |
634 \end{proof} |
631 \end{proof} |
635 |
632 |
636 \subsection{Specializations} |
633 \subsection{Specializations} |
637 \label{sec:specializations} |
634 \label{sec:specializations} |
638 |
635 |
639 The blob complex has two important special cases. |
636 The blob complex has several important special cases. |
640 |
637 |
641 \begin{thm}[Skein modules] |
638 \begin{thm}[Skein modules] |
642 \label{thm:skein-modules} |
639 \label{thm:skein-modules} |
643 \nn{Plain n-categories only?} |
640 \nn{Plain n-categories only?} |
644 The $0$-th blob homology of $X$ is the usual |
641 The $0$-th blob homology of $X$ is the usual |
661 |
658 |
662 Theorem \ref{thm:skein-modules} is immediate from the definition, and |
659 Theorem \ref{thm:skein-modules} is immediate from the definition, and |
663 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 $\bc_*(S^1; -)$. |
660 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 $\bc_*(S^1; -)$. |
664 |
661 |
665 |
662 |
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663 \begin{thm}[Mapping spaces] |
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664 \label{thm:map-recon} |
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665 Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
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666 $B^n \to T$. |
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667 (The case $n=1$ is the usual $A_\infty$-category of paths in $T$.) |
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668 Then |
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669 $$\bc_*(X; \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
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670 \end{thm} |
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671 |
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672 This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. |
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673 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}. |
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674 \todo{sketch proof} |
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675 |
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676 |
666 \subsection{Structure of the blob complex} |
677 \subsection{Structure of the blob complex} |
667 \label{sec:structure} |
678 \label{sec:structure} |
668 |
679 |
669 In the following $\CH{X} = C_*(\Homeo(X))$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. |
680 In the following $\CH{X} = C_*(\Homeo(X))$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. |
670 |
681 |
697 \CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor e_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{e_X} \\ |
708 \CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor e_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{e_X} \\ |
698 \CH{X} \tensor \bc_*(X) \ar[r]^{e_X} & \bc_*(X) |
709 \CH{X} \tensor \bc_*(X) \ar[r]^{e_X} & \bc_*(X) |
699 } |
710 } |
700 \end{equation*} |
711 \end{equation*} |
701 \end{thm} |
712 \end{thm} |
702 |
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703 \nn{if we need to save space, I think this next paragraph could be cut} |
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704 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
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705 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
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706 for any homeomorphic pair $X$ and $Y$, |
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707 satisfying corresponding conditions. |
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708 |
713 |
709 \begin{proof}(Sketch.) |
714 \begin{proof}(Sketch.) |
710 The most convenient way to prove this is to introduce yet another homotopy equivalent version of |
715 The most convenient way to prove this is to introduce yet another homotopy equivalent version of |
711 the blob complex, $\cB\cT_*(X)$. |
716 the blob complex, $\cB\cT_*(X)$. |
712 Blob diagrams have a natural topology, which is ignored by $\bc_*(X)$. |
717 Blob diagrams have a natural topology, which is ignored by $\bc_*(X)$. |
772 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
777 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
773 (see \cite[\S7.1]{1009.5025}). |
778 (see \cite[\S7.1]{1009.5025}). |
774 |
779 |
775 \nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.} |
780 \nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.} |
776 |
781 |
777 \section{Applications} |
782 \section{Higher Deligne conjecture} |
778 \label{sec:applications} |
783 \label{sec:applications} |
779 Finally, we give two applications of the above machinery. |
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780 |
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781 \begin{thm}[Mapping spaces] |
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782 \label{thm:map-recon} |
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783 Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
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784 $B^n \to T$. |
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785 (The case $n=1$ is the usual $A_\infty$-category of paths in $T$.) |
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786 Then |
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787 $$\bc_*(X; \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
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788 \end{thm} |
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789 |
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790 This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. |
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791 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}. |
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792 \todo{sketch proof} |
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793 |
784 |
794 \begin{thm}[Higher dimensional Deligne conjecture] |
785 \begin{thm}[Higher dimensional Deligne conjecture] |
795 \label{thm:deligne} |
786 \label{thm:deligne} |
796 The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. |
787 The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. |
797 Since the little $n{+}1$-balls operad is a suboperad of the $n$-SC operad, |
788 Since the little $n{+}1$-balls operad is a suboperad of the $n$-SC operad, |
843 Kevin Costello, |
834 Kevin Costello, |
844 Mike Freedman, |
835 Mike Freedman, |
845 Justin Roberts, |
836 Justin Roberts, |
846 and |
837 and |
847 Peter Teichner. |
838 Peter Teichner. |
848 \nn{not full list from big paper, but only most significant names} |
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849 We also thank the Aspen Center for Physics for providing a pleasant and productive |
839 We also thank the Aspen Center for Physics for providing a pleasant and productive |
850 environment during the last stages of this project. |
840 environment during the last stages of this project. |
851 \end{acknowledgments} |
841 \end{acknowledgments} |
852 |
842 |
853 %% PNAS does not support submission of supporting .tex files such as BibTeX. |
843 %% PNAS does not support submission of supporting .tex files such as BibTeX. |