197 identity morphisms, and special behavior in dimension $n$ (e.g. enrichment |
197 identity morphisms, and special behavior in dimension $n$ (e.g. enrichment |
198 in some auxiliary category, or strict associativity instead of weak associativity). |
198 in some auxiliary category, or strict associativity instead of weak associativity). |
199 We will treat each of these in turn. |
199 We will treat each of these in turn. |
200 |
200 |
201 To motivate our morphism axiom, consider the venerable notion of the Moore loop space |
201 To motivate our morphism axiom, consider the venerable notion of the Moore loop space |
202 \nn{need citation -- \S 2.2 of Adams' ``Infinite Loop Spaces''?}. |
202 \cite[\S 2.2]{MR505692}. |
203 In the standard definition of a loop space, loops are always parameterized by the unit interval $I = [0,1]$, |
203 In the standard definition of a loop space, loops are always parameterized by the unit interval $I = [0,1]$, |
204 so composition of loops requires a reparameterization $I\cup I \cong I$, and this leads to a proliferation |
204 so composition of loops requires a reparameterization $I\cup I \cong I$, and this leads to a proliferation |
205 of higher associativity relations. |
205 of higher associativity relations. |
206 While this proliferation is manageable for 1-categories (and indeed leads to an elegant theory |
206 While this proliferation is manageable for 1-categories (and indeed leads to an elegant theory |
207 of Stasheff polyhedra and $A_\infty$ categories), it becomes undesirably complex for higher categories. |
207 of Stasheff polyhedra and $A_\infty$ categories), it becomes undesirably complex for higher categories. |
446 |
446 |
447 We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$ |
447 We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$ |
448 with coefficients in the $n$-category $\cC$ as the homotopy colimit along $\cell(W)$ |
448 with coefficients in the $n$-category $\cC$ as the homotopy colimit along $\cell(W)$ |
449 of the functor $\psi_{\cC; W}$ described above. We write this as $\clh{\cC}(W)$. |
449 of the functor $\psi_{\cC; W}$ described above. We write this as $\clh{\cC}(W)$. |
450 |
450 |
451 An explicit realization of the homotopy colimit is provided by the simplices of the functor $\psi_{\cC; W}$. That is, $$\clh{\cC}(W) = \DirectSum_{x} \psi_{\cC; W}(x_0)[m],$$ where $x = x_0 \leq \cdots \leq x_m$ is a simplex in $\cell(W)$. The differential acts on \todo{finish this} |
451 An explicit realization of the homotopy colimit is provided by the simplices of the functor $\psi_{\cC; W}$. That is, $$\clh{\cC}(W) = \DirectSum_{\bar{x}} \psi_{\cC; W}(x_0)[m],$$ where $\bar{x} = x_0 \leq \cdots \leq x_m$ is a simplex in $\cell(W)$. The differential acts on $(\bar{x},a)$ (here $a \in \psi_{\cC; W}(x_0)$) as |
452 |
452 $$\bdy (\bar{x},a) = (\bar{x}, \bdy a) + (-1)^{\deg a} \left( (d_0 \bar{x}, g(a)) + \sum_{i=1}^m (-1)^i (d_i \bar{x}, a) \right)$$ |
453 Alternatively, we can take advantage of the product structure on $\cell(W)$ to realize the homotopy colimit as the cone-product polyhedra of the functor $\psi_{\cC;W}$. (A cone-product polyhedra is obtained from a point by successively taking the cone or taking the product with another cone-product polyhedron.) A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. |
453 where $g$ is the gluing map from $x_0$ to $x_1$, and $d_i \bar{x}$ denotes the $i$-th face of the simplex $\bar{x}$. |
|
454 |
|
455 Alternatively, we can take advantage of the product structure on $\cell(W)$ to realize the homotopy colimit as the cone-product polyhedra of the functor $\psi_{\cC;W}$. (A cone-product polyhedra is obtained from a point by successively taking the cone or taking the product with another cone-product polyhedron; just as simplices correspond to linear graphs, cone-product polyheda correspond to directed trees.) A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. |
454 |
456 |
455 When $\cC$ is a topological $n$-category, |
457 When $\cC$ is a topological $n$-category, |
456 the flexibility available in the construction of a homotopy colimit allows |
458 the flexibility available in the construction of a homotopy colimit allows |
457 us to give a much more explicit description of the blob complex. We'll write $\bc_*(W; \cC)$ for this more explicit version. |
459 us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. |
458 |
460 |
459 We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible} |
461 We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible} |
460 if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that |
462 if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that |
461 each $B_i$ appears as a connected component of one of the $M_j$. Note that this allows the balls to be pairwise either disjoint or nested. Such a collection of balls cuts $W$ into pieces, the connected components of $W \setminus \bigcup \bdy B_i$. These pieces need not be manifolds, but they do automatically have permissible decompositions. |
463 each $B_i$ appears as a connected component of one of the $M_j$. Note that this allows the balls to be pairwise either disjoint or nested. Such a collection of balls cuts $W$ into pieces, the connected components of $W \setminus \bigcup \bdy B_i$. These pieces need not be manifolds, but they do automatically have permissible decompositions. |
462 |
464 |
547 by $\cC$. |
549 by $\cC$. |
548 \begin{equation*} |
550 \begin{equation*} |
549 H_0(\bc_*(X;\cC)) \iso A_{\cC}(X) |
551 H_0(\bc_*(X;\cC)) \iso A_{\cC}(X) |
550 \end{equation*} |
552 \end{equation*} |
551 \end{thm} |
553 \end{thm} |
552 This follows from the fact that the $0$-th homology of a homotopy colimit of \todo{something plain} is the usual colimit, or directly from the explicit description of the blob complex. |
554 This follows from the fact that the $0$-th homology of a homotopy colimit is the usual colimit, or directly from the explicit description of the blob complex. |
553 |
555 |
554 \begin{thm}[Hochschild homology when $X=S^1$] |
556 \begin{thm}[Hochschild homology when $X=S^1$] |
555 \label{thm:hochschild} |
557 \label{thm:hochschild} |
556 The blob complex for a $1$-category $\cC$ on the circle is |
558 The blob complex for a $1$-category $\cC$ on the circle is |
557 quasi-isomorphic to the Hochschild complex. |
559 quasi-isomorphic to the Hochschild complex. |
628 |
630 |
629 \begin{thm}[Product formula] |
631 \begin{thm}[Product formula] |
630 \label{thm:product} |
632 \label{thm:product} |
631 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
633 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
632 Let $\cC$ be an $n$-category. |
634 Let $\cC$ be an $n$-category. |
633 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology. |
635 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ as above. |
634 Then |
636 Then |
635 \[ |
637 \[ |
636 \bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W). |
638 \bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W). |
637 \] |
639 \] |
638 \end{thm} |
640 \end{thm} |
639 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
641 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
640 (see \cite[\S7.1]{1009.5025}). |
642 (see \cite[\S7.1]{1009.5025}). |
641 |
643 |
642 Fix a topological $n$-category $\cC$, which we'll omit from the notation. |
644 Fix a topological $n$-category $\cC$, which we'll now omit from notation. |
643 Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. |
645 Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. |
644 |
646 |
645 \begin{thm}[Gluing formula] |
647 \begin{thm}[Gluing formula] |
646 \label{thm:gluing} |
648 \label{thm:gluing} |
647 \mbox{}% <-- gets the indenting right |
649 \mbox{}% <-- gets the indenting right |
686 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. |
688 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. |
687 |
689 |
688 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. |
690 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. |
689 |
691 |
690 \begin{proof} |
692 \begin{proof} |
691 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 functoriality. |
693 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. |
692 \end{proof} |
694 \end{proof} |
693 |
695 |
694 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 |
696 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 |
695 \[ |
697 \[ |
696 C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}} |
698 C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}} |