378 \] |
377 \] |
379 \item |
378 \item |
380 Product morphisms are compatible with gluing. |
379 Product morphisms are compatible with gluing. |
381 Let $\pi:E\to X$, $\pi_1:E_1\to X_1$, and $\pi_2:E_2\to X_2$ |
380 Let $\pi:E\to X$, $\pi_1:E_1\to X_1$, and $\pi_2:E_2\to X_2$ |
382 be pinched products with $E = E_1\cup E_2$. |
381 be pinched products with $E = E_1\cup E_2$. |
383 Let $a\in \cC(X)$, and let $a_i$ denote the restriction of $a$ to $X_i\sub X$. |
382 Let $a\in \cC(X)$, and let $a_i$ denote the restriction of $a$ to $X_i\subset X$. |
384 Then |
383 Then |
385 \[ |
384 \[ |
386 \pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) . |
385 \pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) . |
387 \] |
386 \] |
388 \item |
387 \item |
405 \end{enumerate} |
404 \end{enumerate} |
406 } %%% end \noop %%% |
405 } %%% end \noop %%% |
407 \end{axiom} |
406 \end{axiom} |
408 |
407 |
409 To state the next axiom we need the notion of {\it collar maps} on $k$-morphisms. |
408 To state the next axiom we need the notion of {\it collar maps} on $k$-morphisms. |
410 Let $X$ be a $k$-ball and $Y\sub\bd X$ be a $(k{-}1)$-ball. |
409 Let $X$ be a $k$-ball and $Y\subset\bd X$ be a $(k{-}1)$-ball. |
411 Let $J$ be a 1-ball. |
410 Let $J$ be a 1-ball. |
412 Let $Y\times_p J$ denote $Y\times J$ pinched along $(\bd Y)\times J$. |
411 Let $Y\times_p J$ denote $Y\times J$ pinched along $(\bd Y)\times J$. |
413 A collar map is an instance of the composition |
412 A collar map is an instance of the composition |
414 \[ |
413 \[ |
415 \cC(X) \to \cC(X\cup_Y (Y\times_p J)) \to \cC(X) , |
414 \cC(X) \to \cC(X\cup_Y (Y\times_p J)) \to \cC(X) , |
438 |
437 |
439 \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
438 \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
440 \label{axiom:families} |
439 \label{axiom:families} |
441 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
440 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
442 \[ |
441 \[ |
443 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
442 C_*(\Homeo_\bd(X))\tensor \cC(X; c) \to \cC(X; c) . |
444 \] |
443 \] |
445 These action maps are required to be associative up to homotopy, |
444 These action maps are required to be associative up to homotopy, |
446 and also compatible with composition (gluing) in the sense that |
445 and also compatible with composition (gluing) in the sense that |
447 a diagram like the one in Theorem \ref{thm:CH} commutes. |
446 a diagram like the one in Theorem \ref{thm:CH} commutes. |
448 \end{axiom} |
447 \end{axiom} |
467 that is, certain cell complexes embedded in $X$, with the codimension-$j$ cells labeled by $j$-morphisms of $C$. |
466 that is, certain cell complexes embedded in $X$, with the codimension-$j$ cells labeled by $j$-morphisms of $C$. |
468 If $X$ is an $n$-ball, identify two such string diagrams if they evaluate to the same $n$-morphism of $C$. |
467 If $X$ is an $n$-ball, identify two such string diagrams if they evaluate to the same $n$-morphism of $C$. |
469 Boundary restrictions and gluing are again straightforward to define. |
468 Boundary restrictions and gluing are again straightforward to define. |
470 Define product morphisms via product cell decompositions. |
469 Define product morphisms via product cell decompositions. |
471 |
470 |
472 |
471 \subsection{Example (bordism)} |
473 \nn{also do bordism category} |
472 When $X$ is a $k$-ball with $k<n$, $\Bord^n(X)$ is the set of all $k$-dimensional |
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473 submanifolds $W$ in $X\times \bbR^\infty$ which project to $X$ transversely |
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474 to $\bd X$. |
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475 For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes rel boundary of such $n$-dimensional submanifolds. |
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476 |
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477 There is an $A_\infty$ analogue enriched in topological spaces, where at the top level we take all such submanifolds, rather than homeomorphism classes. For each fixed $\bdy W \subset \bdy X \times \bbR^\infty$, we can topologize the set of submanifolds by ambient isotopy rel boundary. |
474 |
478 |
475 \subsection{The blob complex} |
479 \subsection{The blob complex} |
476 \subsubsection{Decompositions of manifolds} |
480 \subsubsection{Decompositions of manifolds} |
477 |
481 |
478 A \emph{ball decomposition} of $W$ is a |
482 A \emph{ball decomposition} of $W$ is a |
501 |
505 |
502 An $n$-category $\cC$ determines |
506 An $n$-category $\cC$ determines |
503 a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
507 a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
504 (possibly with additional structure if $k=n$). |
508 (possibly with additional structure if $k=n$). |
505 Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
509 Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
506 and there is a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries |
510 and there is a subset $\cC(X)\spl \subset \cC(X)$ of morphisms whose boundaries |
507 are splittable along this decomposition. |
511 are splittable along this decomposition. |
508 |
512 |
509 \begin{defn} |
513 \begin{defn} |
510 Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
514 Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
511 For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset |
515 For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset |
512 \begin{equation*} |
516 \begin{equation*} |
513 %\label{eq:psi-C} |
517 %\label{eq:psi-C} |
514 \psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl |
518 \psi_{\cC;W}(x) \subset \prod_a \cC(X_a)\spl |
515 \end{equation*} |
519 \end{equation*} |
516 where the restrictions to the various pieces of shared boundaries amongst the cells |
520 where the restrictions to the various pieces of shared boundaries amongst the cells |
517 $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells). When $k=n$, the `subset' and `product' in the above formula should be interpreted in the appropriate enriching category. |
521 $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells). When $k=n$, the `subset' and `product' in the above formula should be interpreted in the appropriate enriching category. |
518 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$. |
522 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$. |
519 \end{defn} |
523 \end{defn} |
677 \item For |
681 \item For |
678 any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
682 any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
679 (using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy). |
683 (using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy). |
680 \begin{equation*} |
684 \begin{equation*} |
681 \xymatrix@C+0.3cm{ |
685 \xymatrix@C+0.3cm{ |
682 \CH{X} \otimes \bc_*(X) |
686 \CH{X} \tensor \bc_*(X) |
683 \ar[r]_{e_{X}} \ar[d]^{\gl^{\Homeo}_Y \otimes \gl_Y} & |
687 \ar[r]_{e_{X}} \ar[d]^{\gl^{\Homeo}_Y \tensor \gl_Y} & |
684 \bc_*(X) \ar[d]_{\gl_Y} \\ |
688 \bc_*(X) \ar[d]_{\gl_Y} \\ |
685 \CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]_<<<<<<<{e_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow) |
689 \CH{X \bigcup_Y \selfarrow} \tensor \bc_*(X \bigcup_Y \selfarrow) \ar[r]_<<<<<<<{e_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow) |
686 } |
690 } |
687 \end{equation*} |
691 \end{equation*} |
688 \end{enumerate} |
692 \end{enumerate} |
689 |
693 |
690 Further, this map is associative, in the sense that the following diagram commutes (up to homotopy). |
694 Further, this map is associative, in the sense that the following diagram commutes (up to homotopy). |
810 The little disks operad $LD$ is homotopy equivalent to |
814 The little disks operad $LD$ is homotopy equivalent to |
811 \nn{suboperad of} |
815 \nn{suboperad of} |
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 $Hoch^*(C, C)$. |
816 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 $Hoch^*(C, C)$. |
813 The usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923}) gives a map |
817 The usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923}) gives a map |
814 \[ |
818 \[ |
815 C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}} |
819 C_*(LD_k)\tensor \overbrace{Hoch^*(C, C)\tensor\cdots\tensor Hoch^*(C, C)}^{\text{$k$ copies}} |
816 \to Hoch^*(C, C), |
820 \to Hoch^*(C, C), |
817 \] |
821 \] |
818 which we now see to be a specialization of Theorem \ref{thm:deligne}. |
822 which we now see to be a specialization of Theorem \ref{thm:deligne}. |
819 |
823 |
820 |
824 |