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136 \begin{article} |
136 \begin{article} |
137 |
137 |
138 \begin{abstract} |
138 \begin{abstract} |
|
139 \nn{needs revision} |
139 We explain the need for new axioms for topological quantum field theories that include ideas from derived |
140 We explain the need for new axioms for topological quantum field theories that include ideas from derived |
140 categories and homotopy theory. We summarize our axioms for higher categories, and describe the ``blob complex". |
141 categories and homotopy theory. We summarize our axioms for higher categories, and describe the ``blob complex". |
141 Fixing an $n$-category $\cC$, the blob complex associates a chain complex $\bc_*(W;\cC)$ to any $n$-manifold $W$. |
142 Fixing an $n$-category $\cC$, the blob complex associates a chain complex $\bc_*(W;\cC)$ to any $n$-manifold $W$. |
142 The $0$-th homology of this chain complex recovers the usual TQFT invariants of $W$. |
143 The $0$-th homology of this chain complex recovers the usual TQFT invariants of $W$. |
143 The higher homology groups should be viewed as generalizations of Hochschild homology. |
144 The higher homology groups should be viewed as generalizations of Hochschild homology. |
234 in a concrete version of the homotopy colimit.) |
235 in a concrete version of the homotopy colimit.) |
235 We then review some basic properties of the blob complex, and finish by showing how it |
236 We then review some basic properties of the blob complex, and finish by showing how it |
236 yields a higher categorical and higher dimensional generalization of Deligne's |
237 yields a higher categorical and higher dimensional generalization of Deligne's |
237 conjecture on Hochschild cochains and the little 2-disks operad. |
238 conjecture on Hochschild cochains and the little 2-disks operad. |
238 |
239 |
|
240 \nn{needs revision} |
239 Of course, there are currently many interesting alternative notions of $n$-category and of TQFT. |
241 Of course, there are currently many interesting alternative notions of $n$-category and of TQFT. |
240 We note that our $n$-categories are both more and less general |
242 We note that our $n$-categories are both more and less general |
241 than the ``fully dualizable" ones which play a prominent role in \cite{0905.0465}. |
243 than the ``fully dualizable" ones which play a prominent role in \cite{0905.0465}. |
242 They are more general in that we make no duality assumptions in the top dimension $n{+}1$. |
244 They are more general in that we make no duality assumptions in the top dimension $n{+}1$. |
243 They are less general in that we impose stronger duality requirements in dimensions 0 through $n$. |
245 They are less general in that we impose stronger duality requirements in dimensions 0 through $n$. |
269 %of the main theorems; and (2) specify a minimal set of generators and/or axioms. |
271 %of the main theorems; and (2) specify a minimal set of generators and/or axioms. |
270 %We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second. |
272 %We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second. |
271 %More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.} |
273 %More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.} |
272 |
274 |
273 We will define two variations simultaneously, as all but one of the axioms are identical in the two cases. |
275 We will define two variations simultaneously, as all but one of the axioms are identical in the two cases. |
274 These variations are ``isotopy $n$-categories", where homeomorphisms fixing the boundary |
276 These variations are ``plain $n$-categories", where homeomorphisms fixing the boundary |
275 act trivially on the sets associated to $n$-balls |
277 act trivially on the sets associated to $n$-balls |
276 (and these sets are usually vector spaces or more generally modules over a commutative ring) |
278 (and these sets are usually vector spaces or more generally modules over a commutative ring) |
277 and ``$A_\infty$ $n$-categories", where there is a homotopy action of |
279 and ``$A_\infty$ $n$-categories", where there is a homotopy action of |
278 $k$-parameter families of homeomorphisms on these sets |
280 $k$-parameter families of homeomorphisms on these sets |
279 (which are usually chain complexes or topological spaces). |
281 (which are usually chain complexes or topological spaces). |
374 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
376 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
375 to the intersection of the boundaries of $B$ and $B_i$. |
377 to the intersection of the boundaries of $B$ and $B_i$. |
376 If $k < n$, |
378 If $k < n$, |
377 or if $k=n$ and we are in the $A_\infty$ case, |
379 or if $k=n$ and we are in the $A_\infty$ case, |
378 we require that $\gl_Y$ is injective. |
380 we require that $\gl_Y$ is injective. |
379 (For $k=n$ in the isotopy $n$-category case, see Axiom \ref{axiom:extended-isotopies}.) |
381 (For $k=n$ in the plain $n$-category case, see Axiom \ref{axiom:extended-isotopies}.) |
380 \end{axiom} |
382 \end{axiom} |
381 |
383 |
382 \begin{axiom}[Strict associativity] \label{nca-assoc}\label{axiom:associativity} |
384 \begin{axiom}[Strict associativity] \label{nca-assoc}\label{axiom:associativity} |
383 The gluing maps above are strictly associative. |
385 The gluing maps above are strictly associative. |
384 Given any decomposition of a ball $B$ into smaller balls |
386 Given any decomposition of a ball $B$ into smaller balls |
460 where the first arrow is gluing with a product morphism on $Y\times_p J$ and |
462 where the first arrow is gluing with a product morphism on $Y\times_p J$ and |
461 the second is induced by a homeomorphism from $X\cup_Y (Y\times_p J)$ to $X$ which restricts |
463 the second is induced by a homeomorphism from $X\cup_Y (Y\times_p J)$ to $X$ which restricts |
462 to the identity on the boundary. |
464 to the identity on the boundary. |
463 |
465 |
464 |
466 |
465 \begin{axiom}[\textup{\textbf{[for isotopy $n$-categories]}} Extended isotopy invariance in dimension $n$.] |
467 \begin{axiom}[\textup{\textbf{[for plain $n$-categories]}} Extended isotopy invariance in dimension $n$.] |
466 \label{axiom:extended-isotopies} |
468 \label{axiom:extended-isotopies} |
467 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
469 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
468 to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
470 to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
469 Then $f$ acts trivially on $\cC(X)$. |
471 Then $f$ acts trivially on $\cC(X)$. |
470 In addition, collar maps act trivially on $\cC(X)$. |
472 In addition, collar maps act trivially on $\cC(X)$. |
564 \begin{equation*} |
566 \begin{equation*} |
565 %\label{eq:psi-C} |
567 %\label{eq:psi-C} |
566 \psi_{\cC;W}(x) \subset \prod_a \cC(X_a)\spl |
568 \psi_{\cC;W}(x) \subset \prod_a \cC(X_a)\spl |
567 \end{equation*} |
569 \end{equation*} |
568 where the restrictions to the various pieces of shared boundaries amongst the balls |
570 where the restrictions to the various pieces of shared boundaries amongst the balls |
569 $X_a$ all agree (this is a fibered product of all the labels of $k$-balls over the labels of $k-1$-manifolds). |
571 $X_a$ all agree (similar to a fibered product). |
570 When $k=n$, the ``subset" and ``product" in the above formula should be |
572 When $k=n$, the ``subset" and ``product" in the above formula should be |
571 interpreted in the appropriate enriching category. |
573 interpreted in the appropriate enriching category. |
572 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 $\cC$. |
574 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 $\cC$. |
573 \end{defn} |
575 \end{defn} |
574 |
576 |
575 We will use the term ``field on $W$" to refer to a point of this functor, |
577 %We will use the term ``field on $W$" to refer to a point of this functor, |
576 that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$. |
578 %that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$. |
577 |
579 |
578 |
580 |
579 \subsubsection{Colimits} |
581 \subsubsection{Colimits} |
580 Recall that our definition of an $n$-category is essentially a collection of functors |
582 Recall that our definition of an $n$-category is essentially a collection of functors |
581 defined on the categories of homeomorphisms of $k$-balls |
583 defined on the categories of homeomorphisms of $k$-balls |
583 It is natural to hope to extend such functors to the |
585 It is natural to hope to extend such functors to the |
584 larger categories of all $k$-manifolds (again, with homeomorphisms). |
586 larger categories of all $k$-manifolds (again, with homeomorphisms). |
585 In fact, the axioms stated above already require such an extension to $k$-spheres for $k<n$. |
587 In fact, the axioms stated above already require such an extension to $k$-spheres for $k<n$. |
586 |
588 |
587 The natural construction achieving this is a colimit along the poset of permissible decompositions. |
589 The natural construction achieving this is a colimit along the poset of permissible decompositions. |
588 Given an isotopy $n$-category $\cC$, |
590 Given a plain $n$-category $\cC$, |
589 we will denote its extension to all manifolds by $\cl{\cC}$. On a $k$-manifold $W$, with $k \leq n$, |
591 we will denote its extension to all manifolds by $\cl{\cC}$. On a $k$-manifold $W$, with $k \leq n$, |
590 this is defined to be the colimit along $\cell(W)$ of the functor $\psi_{\cC;W}$. |
592 this is defined to be the colimit along $\cell(W)$ of the functor $\psi_{\cC;W}$. |
591 Note that Axioms \ref{axiom:composition} and \ref{axiom:associativity} |
593 Note that Axioms \ref{axiom:composition} and \ref{axiom:associativity} |
592 imply that $\cl{\cC}(X) \iso \cC(X)$ when $X$ is a $k$-ball with $k<n$. |
594 imply that $\cl{\cC}(X) \iso \cC(X)$ when $X$ is a $k$-ball with $k<n$. |
593 Suppose that $\cC$ is enriched in vector spaces: this means that given boundary conditions $c \in \cl{\cC}(\bdy X)$, for $X$ an $n$-ball, |
595 Suppose that $\cC$ is enriched in vector spaces: this means that given boundary conditions $c \in \cl{\cC}(\bdy X)$, for $X$ an $n$-ball, |
620 |
622 |
621 %When $\cC$ is a topological $n$-category, |
623 %When $\cC$ is a topological $n$-category, |
622 %the flexibility available in the construction of a homotopy colimit allows |
624 %the flexibility available in the construction of a homotopy colimit allows |
623 %us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. |
625 %us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. |
624 %\todo{either need to explain why this is the same, or significantly rewrite this section} |
626 %\todo{either need to explain why this is the same, or significantly rewrite this section} |
625 When $\cC$ is the isotopy $n$-category based on string diagrams for a traditional |
627 When $\cC$ is the plain $n$-category based on string diagrams for a traditional |
626 $n$-category $C$, |
628 $n$-category $C$, |
627 one can show \cite{1009.5025} that the above two constructions of the homotopy colimit |
629 one can show \cite{1009.5025} that the above two constructions of the homotopy colimit |
628 are equivalent to the more concrete construction which we describe next, and which we denote $\bc_*(W; \cC)$. |
630 are equivalent to the more concrete construction which we describe next, and which we denote $\bc_*(W; \cC)$. |
629 Roughly speaking, the generators of $\bc_k(W; \cC)$ are string diagrams on $W$ together with |
631 Roughly speaking, the generators of $\bc_k(W; \cC)$ are string diagrams on $W$ together with |
630 a configuration of $k$ balls (or ``blobs") in $W$ whose interiors are pairwise disjoint or nested. |
632 a configuration of $k$ balls (or ``blobs") in $W$ whose interiors are pairwise disjoint or nested. |
637 each $B_i$ appears as a connected component of one of the $M_j$. |
639 each $B_i$ appears as a connected component of one of the $M_j$. |
638 Note that this forces the balls to be pairwise either disjoint or nested. |
640 Note that this forces the balls to be pairwise either disjoint or nested. |
639 Such a collection of balls cuts $W$ into pieces, the connected components of $W \setminus \bigcup \bdy B_i$. |
641 Such a collection of balls cuts $W$ into pieces, the connected components of $W \setminus \bigcup \bdy B_i$. |
640 These pieces need not be manifolds, but they do automatically have permissible decompositions. |
642 These pieces need not be manifolds, but they do automatically have permissible decompositions. |
641 |
643 |
642 The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams. A $k$-blob diagram consists of |
644 The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams. |
|
645 A $k$-blob diagram consists of |
643 \begin{itemize} |
646 \begin{itemize} |
644 \item a permissible collection of $k$ embedded balls, and |
647 \item a permissible collection of $k$ embedded balls, and |
645 \item for each resulting piece of $W$, a field, |
648 \item for each resulting piece of $W$, a field, |
646 \end{itemize} |
649 \end{itemize} |
647 such that for any innermost blob $B$, the field on $B$ goes to zero under the gluing map from $\cC$. |
650 such that for any innermost blob $B$, the field on $B$ goes to zero under the gluing map from $\cC$. |
648 We call such a field a ``null field on $B$". |
651 We call such a field a ``null field on $B$". |
649 |
652 |
650 The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with alternating signs. |
653 The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with alternating signs. |
651 |
654 |
652 We now spell this out for some small values of $k$. For $k=0$, the $0$-blob group is simply fields on $W$. |
655 We now spell this out for some small values of $k$. |
|
656 For $k=0$, the $0$-blob group is simply fields on $W$. |
653 For $k=1$, a generator consists of a field on $W$ and a ball, such that the restriction of the field to that ball is a null field. |
657 For $k=1$, a generator consists of a field on $W$ and a ball, such that the restriction of the field to that ball is a null field. |
654 The differential simply forgets the ball. |
658 The differential simply forgets the ball. |
655 Thus we see that $H_0$ of the blob complex is the quotient of fields by fields which are null on some ball. |
659 Thus we see that $H_0$ of the blob complex is the quotient of fields by fields which are null on some ball. |
656 |
660 |
657 For $k=2$, we have a two types of generators; they each consists of a field $f$ on $W$, and two balls $B_1$ and $B_2$. |
661 For $k=2$, we have a two types of generators; they each consists of a field $f$ on $W$, and two balls $B_1$ and $B_2$. |
728 |
732 |
729 The blob complex has several important special cases. |
733 The blob complex has several important special cases. |
730 |
734 |
731 \begin{thm}[Skein modules] |
735 \begin{thm}[Skein modules] |
732 \label{thm:skein-modules} |
736 \label{thm:skein-modules} |
733 Suppose $\cC$ is an isotopy $n$-category. |
737 Suppose $\cC$ is a plain $n$-category. |
734 The $0$-th blob homology of $X$ is the usual |
738 The $0$-th blob homology of $X$ is the usual |
735 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
739 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
736 by $\cC$. |
740 by $\cC$. |
737 \begin{equation*} |
741 \begin{equation*} |
738 H_0(\bc_*(X;\cC)) \iso \cl{\cC}(X) |
742 H_0(\bc_*(X;\cC)) \iso \cl{\cC}(X) |
872 blob complexes for the $A_\infty$ $n$-categories constructed as above. |
876 blob complexes for the $A_\infty$ $n$-categories constructed as above. |
873 |
877 |
874 \begin{thm}[Product formula] |
878 \begin{thm}[Product formula] |
875 \label{thm:product} |
879 \label{thm:product} |
876 Let $W$ be a $k$-manifold and $Y$ be an $n{-}k$ manifold. |
880 Let $W$ be a $k$-manifold and $Y$ be an $n{-}k$ manifold. |
877 Let $\cC$ be an isotopy $n$-category. |
881 Let $\cC$ be a plain $n$-category. |
878 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ as above. |
882 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ as above. |
879 Then |
883 Then |
880 \[ |
884 \[ |
881 \bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W). |
885 \bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W). |
882 \] |
886 \] |
902 A standard acyclic models argument now constructs the homotopy inverse. |
906 A standard acyclic models argument now constructs the homotopy inverse. |
903 \end{proof} |
907 \end{proof} |
904 |
908 |
905 %\nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.} |
909 %\nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.} |
906 |
910 |
907 \section{Deligne's conjecture for $n$-categories} |
911 \section{Extending Deligne's conjecture to $n$-categories} |
908 \label{sec:applications} |
912 \label{sec:applications} |
909 |
913 |
910 Let $M$ and $N$ be $n$-manifolds with common boundary $E$. |
914 Let $M$ and $N$ be $n$-manifolds with common boundary $E$. |
911 Recall (Theorem \ref{thm:gluing}) that the $A_\infty$ category $A = \bc_*(E)$ |
915 Recall (Theorem \ref{thm:gluing}) that the $A_\infty$ category $A = \bc_*(E)$ |
912 acts on $\bc_*(M)$ and $\bc_*(N)$. |
916 acts on $\bc_*(M)$ and $\bc_*(N)$. |
950 which satisfy the operad compatibility conditions. |
954 which satisfy the operad compatibility conditions. |
951 |
955 |
952 \begin{proof} (Sketch.) |
956 \begin{proof} (Sketch.) |
953 We have already defined the action of mapping cylinders, in Theorem \ref{thm:evaluation}, |
957 We have already defined the action of mapping cylinders, in Theorem \ref{thm:evaluation}, |
954 and the action of surgeries is just composition of maps of $A_\infty$-modules. |
958 and the action of surgeries is just composition of maps of $A_\infty$-modules. |
955 We only need to check that the relations of the surgery cylinded operad are satisfied. |
959 We only need to check that the relations of the surgery cylinder operad are satisfied. |
956 This follows from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and associativity. |
960 This follows from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and associativity. |
957 \end{proof} |
961 \end{proof} |
958 |
962 |
959 Consider the special case where $n=1$ and all of the $M_i$'s and $N_i$'s are intervals. |
963 Consider the special case where $n=1$ and all of the $M_i$'s and $N_i$'s are intervals. |
960 We have that $SC^1_{\overline{M}, \overline{N}}$ is homotopy equivalent to the little |
964 We have that $SC^1_{\overline{M}, \overline{N}}$ is homotopy equivalent to the little |