372 \] |
371 \] |
373 \item |
372 \item |
374 Product morphisms are compatible with gluing. |
373 Product morphisms are compatible with gluing. |
375 Let $\pi:E\to X$, $\pi_1:E_1\to X_1$, and $\pi_2:E_2\to X_2$ |
374 Let $\pi:E\to X$, $\pi_1:E_1\to X_1$, and $\pi_2:E_2\to X_2$ |
376 be pinched products with $E = E_1\cup E_2$. |
375 be pinched products with $E = E_1\cup E_2$. |
377 Let $a\in \cC(X)$, and let $a_i$ denote the restriction of $a$ to $X_i\sub X$. |
376 Let $a\in \cC(X)$, and let $a_i$ denote the restriction of $a$ to $X_i\subset X$. |
378 Then |
377 Then |
379 \[ |
378 \[ |
380 \pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) . |
379 \pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) . |
381 \] |
380 \] |
382 \item |
381 \item |
399 \end{enumerate} |
398 \end{enumerate} |
400 } %%% end \noop %%% |
399 } %%% end \noop %%% |
401 \end{axiom} |
400 \end{axiom} |
402 |
401 |
403 To state the next axiom we need the notion of {\it collar maps} on $k$-morphisms. |
402 To state the next axiom we need the notion of {\it collar maps} on $k$-morphisms. |
404 Let $X$ be a $k$-ball and $Y\sub\bd X$ be a $(k{-}1)$-ball. |
403 Let $X$ be a $k$-ball and $Y\subset\bd X$ be a $(k{-}1)$-ball. |
405 Let $J$ be a 1-ball. |
404 Let $J$ be a 1-ball. |
406 Let $Y\times_p J$ denote $Y\times J$ pinched along $(\bd Y)\times J$. |
405 Let $Y\times_p J$ denote $Y\times J$ pinched along $(\bd Y)\times J$. |
407 A collar map is an instance of the composition |
406 A collar map is an instance of the composition |
408 \[ |
407 \[ |
409 \cC(X) \to \cC(X\cup_Y (Y\times_p J)) \to \cC(X) , |
408 \cC(X) \to \cC(X\cup_Y (Y\times_p J)) \to \cC(X) , |
432 |
431 |
433 \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
432 \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
434 \label{axiom:families} |
433 \label{axiom:families} |
435 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
434 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
436 \[ |
435 \[ |
437 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
436 C_*(\Homeo_\bd(X))\tensor \cC(X; c) \to \cC(X; c) . |
438 \] |
437 \] |
439 These action maps are required to be associative up to homotopy, |
438 These action maps are required to be associative up to homotopy, |
440 and also compatible with composition (gluing) in the sense that |
439 and also compatible with composition (gluing) in the sense that |
441 a diagram like the one in Theorem \ref{thm:CH} commutes. |
440 a diagram like the one in Theorem \ref{thm:CH} commutes. |
442 \end{axiom} |
441 \end{axiom} |
462 If $X$ is an $n$-ball, identify two such string diagrams if they evaluate to the same $n$-morphism of $C$. |
461 If $X$ is an $n$-ball, identify two such string diagrams if they evaluate to the same $n$-morphism of $C$. |
463 Boundary restrictions and gluing are again straightforward to define. |
462 Boundary restrictions and gluing are again straightforward to define. |
464 Define product morphisms via product cell decompositions. |
463 Define product morphisms via product cell decompositions. |
465 |
464 |
466 |
465 |
467 \nn{also do bordism category} |
|
468 |
466 |
469 \subsection{The blob complex} |
467 \subsection{The blob complex} |
470 \subsubsection{Decompositions of manifolds} |
468 \subsubsection{Decompositions of manifolds} |
471 |
469 |
472 A \emph{ball decomposition} of $W$ is a |
470 A \emph{ball decomposition} of $W$ is a |
495 |
493 |
496 An $n$-category $\cC$ determines |
494 An $n$-category $\cC$ determines |
497 a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
495 a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
498 (possibly with additional structure if $k=n$). |
496 (possibly with additional structure if $k=n$). |
499 Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
497 Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
500 and there is a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries |
498 and there is a subset $\cC(X)\spl \subset \cC(X)$ of morphisms whose boundaries |
501 are splittable along this decomposition. |
499 are splittable along this decomposition. |
502 |
500 |
503 \begin{defn} |
501 \begin{defn} |
504 Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
502 Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
505 For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset |
503 For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset |
506 \begin{equation*} |
504 \begin{equation*} |
507 %\label{eq:psi-C} |
505 %\label{eq:psi-C} |
508 \psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl |
506 \psi_{\cC;W}(x) \subset \prod_a \cC(X_a)\spl |
509 \end{equation*} |
507 \end{equation*} |
510 where the restrictions to the various pieces of shared boundaries amongst the cells |
508 where the restrictions to the various pieces of shared boundaries amongst the cells |
511 $X_a$ 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. |
509 $X_a$ 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. |
512 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 $\cC$. |
510 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 $\cC$. |
513 \end{defn} |
511 \end{defn} |
671 \item For |
669 \item For |
672 any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
670 any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
673 (using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy). |
671 (using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy). |
674 \begin{equation*} |
672 \begin{equation*} |
675 \xymatrix@C+0.3cm{ |
673 \xymatrix@C+0.3cm{ |
676 \CH{X} \otimes \bc_*(X) |
674 \CH{X} \tensor \bc_*(X) |
677 \ar[r]_{e_{X}} \ar[d]^{\gl^{\Homeo}_Y \otimes \gl_Y} & |
675 \ar[r]_{e_{X}} \ar[d]^{\gl^{\Homeo}_Y \tensor \gl_Y} & |
678 \bc_*(X) \ar[d]_{\gl_Y} \\ |
676 \bc_*(X) \ar[d]_{\gl_Y} \\ |
679 \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) |
677 \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) |
680 } |
678 } |
681 \end{equation*} |
679 \end{equation*} |
682 \end{enumerate} |
680 \end{enumerate} |
683 |
681 |
684 Futher, this map is associative, in the sense that the following diagram commutes (up to homotopy). |
682 Futher, this map is associative, in the sense that the following diagram commutes (up to homotopy). |
780 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. |
778 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. |
781 \end{proof} |
779 \end{proof} |
782 |
780 |
783 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 |
781 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 |
784 \[ |
782 \[ |
785 C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}} |
783 C_*(LD_k)\tensor \overbrace{Hoch^*(C, C)\tensor\cdots\tensor Hoch^*(C, C)}^{\text{$k$ copies}} |
786 \to Hoch^*(C, C), |
784 \to Hoch^*(C, C), |
787 \] |
785 \] |
788 which we now see to be a specialization of Theorem \ref{thm:deligne}. |
786 which we now see to be a specialization of Theorem \ref{thm:deligne}. |
789 |
787 |
790 |
788 |