222 in a concrete version of the homotopy colimit.) |
222 in a concrete version of the homotopy colimit.) |
223 We then review some basic properties of the blob complex, and finish by showing how it |
223 We then review some basic properties of the blob complex, and finish by showing how it |
224 yields a higher categorical and higher dimensional generalization of Deligne's |
224 yields a higher categorical and higher dimensional generalization of Deligne's |
225 conjecture on Hochschild cochains and the little 2-disks operad. |
225 conjecture on Hochschild cochains and the little 2-disks operad. |
226 |
226 |
227 \nn{maybe this is not necessary?} \nn{let's move this to somewhere later, if we keep it} |
227 Of course, there are currently many interesting alternative notions of $n$-category and of TQFT. |
228 In an attempt to forestall any confusion that might arise from different definitions of |
228 We note that our $n$-categories are both more and less general |
229 ``$n$-category" and ``TQFT", we note that our $n$-categories are both more and less general |
|
230 than the ``fully dualizable" ones which play a prominent role in \cite{0905.0465}. |
229 than the ``fully dualizable" ones which play a prominent role in \cite{0905.0465}. |
231 More general in that we make no duality assumptions in the top dimension $n+1$. |
230 They are more general in that we make no duality assumptions in the top dimension $n+1$. |
232 Less general in that we impose stronger duality requirements in dimensions 0 through $n$. |
231 They are less general in that we impose stronger duality requirements in dimensions 0 through $n$. |
233 Thus our $n$-categories correspond to $(n{+}\epsilon)$-dimensional unoriented or oriented TQFTs, while |
232 Thus our $n$-categories correspond to $(n{+}\epsilon)$-dimensional unoriented or oriented TQFTs, while |
234 Lurie's (fully dualizable) $n$-categories correspond to $(n{+}1)$-dimensional framed TQFTs. |
233 Lurie's (fully dualizable) $n$-categories correspond to $(n{+}1)$-dimensional framed TQFTs. |
235 |
234 |
236 At several points we only sketch an argument briefly; full details can be found in \cite{1009.5025}. In this paper we attempt to give a clear view of the big picture without getting bogged down in technical details. |
235 At several points we only sketch an argument briefly; full details can be found in \cite{1009.5025}. In this paper we attempt to give a clear view of the big picture without getting bogged down in technical details. |
237 |
236 |
280 Thus we can have the simplicity of strict associativity in exchange for more morphisms. |
279 Thus we can have the simplicity of strict associativity in exchange for more morphisms. |
281 We wish to imitate this strategy in higher categories. |
280 We wish to imitate this strategy in higher categories. |
282 Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with |
281 Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with |
283 a product of $k$ intervals (c.f. \cite{0909.2212}) but rather with any $k$-ball, that is, any $k$-manifold which is homeomorphic |
282 a product of $k$ intervals (c.f. \cite{0909.2212}) but rather with any $k$-ball, that is, any $k$-manifold which is homeomorphic |
284 to the standard $k$-ball $B^k$. |
283 to the standard $k$-ball $B^k$. |
285 \nn{maybe add that in addition we want functoriality} |
|
286 |
284 |
287 By default our balls are unoriented, |
285 By default our balls are unoriented, |
288 but it is useful at times to vary this, |
286 but it is useful at times to vary this, |
289 for example by considering oriented or Spin balls. |
287 for example by considering oriented or Spin balls. |
290 We can also consider more exotic structures, such as balls with a map to some target space, |
288 We can also consider more exotic structures, such as balls with a map to some target space, |
302 homeomorphisms which are not the identity on the boundary of the $k$-ball. |
300 homeomorphisms which are not the identity on the boundary of the $k$-ball. |
303 The action of these homeomorphisms gives the ``strong duality" structure. |
301 The action of these homeomorphisms gives the ``strong duality" structure. |
304 As such, we don't subdivide the boundary of a morphism |
302 As such, we don't subdivide the boundary of a morphism |
305 into domain and range --- the duality operations can convert between domain and range. |
303 into domain and range --- the duality operations can convert between domain and range. |
306 |
304 |
307 Later \nn{make sure this actually happens, or reorganise} we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$ from arbitrary manifolds to sets. We need the restriction of these functors to $k$-spheres, for $k<n$, for the next axiom. |
305 Later we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$ from arbitrary manifolds to sets. We need the restriction of these functors to $k$-spheres, for $k<n$, for the next axiom. |
308 |
306 |
309 \begin{axiom}[Boundaries]\label{nca-boundary} |
307 \begin{axiom}[Boundaries]\label{nca-boundary} |
310 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
308 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
311 These maps, for various $X$, comprise a natural transformation of functors. |
309 These maps, for various $X$, comprise a natural transformation of functors. |
312 \end{axiom} |
310 \end{axiom} |
637 \begin{property}[Contractibility] |
635 \begin{property}[Contractibility] |
638 \label{property:contractibility}% |
636 \label{property:contractibility}% |
639 The blob complex on an $n$-ball is contractible in the sense |
637 The blob complex on an $n$-ball is contractible in the sense |
640 that it is homotopic to its $0$-th homology, and this is just the vector space associated to the ball by the $n$-category. |
638 that it is homotopic to its $0$-th homology, and this is just the vector space associated to the ball by the $n$-category. |
641 \begin{equation*} |
639 \begin{equation*} |
642 \xymatrix{\bc_*(B^n;\cC) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cC)) \ar[r]^(0.6)\iso & \cC(B^n)} |
640 \xymatrix{\bc_*(B^n;\cC) \ar[r]^(0.4){\htpy} & H_0(\bc_*(B^n;\cC)) \ar[r]^(0.6)\iso & \cC(B^n)} |
643 \end{equation*} |
641 \end{equation*} |
644 \end{property} |
642 \end{property} |
645 %\nn{maybe should say something about the $A_\infty$ case} |
643 %\nn{maybe should say something about the $A_\infty$ case} |
646 |
644 |
647 \begin{proof}(Sketch) |
645 \begin{proof}(Sketch) |
649 obtained by adding an outer $(k{+}1)$-st blob consisting of all $B^n$. |
647 obtained by adding an outer $(k{+}1)$-st blob consisting of all $B^n$. |
650 For $k=0$ we choose a splitting $s: H_0(\bc_*(B^n)) \to \bc_0(B^n)$ and send |
648 For $k=0$ we choose a splitting $s: H_0(\bc_*(B^n)) \to \bc_0(B^n)$ and send |
651 $x\in \bc_0(B^n)$ to $x - s([x])$, where $[x]$ denotes the image of $x$ in $H_0(\bc_*(B^n))$. |
649 $x\in \bc_0(B^n)$ to $x - s([x])$, where $[x]$ denotes the image of $x$ in $H_0(\bc_*(B^n))$. |
652 \end{proof} |
650 \end{proof} |
653 |
651 |
654 If $\cC$ is an $A-\infty$ $n$-category then $\bc_*(B^n;\cC)$ is still homotopy equivalent to $\cC(B^n)$, |
652 If $\cC$ is an $A_\infty$ $n$-category then $\bc_*(B^n;\cC)$ is still homotopy equivalent to $\cC(B^n)$, |
655 but this is no longer concentrated in degree zero. |
653 but this is no longer concentrated in degree zero. |
656 |
654 |
657 \subsection{Specializations} |
655 \subsection{Specializations} |
658 \label{sec:specializations} |
656 \label{sec:specializations} |
659 |
657 |
660 The blob complex has several important special cases. |
658 The blob complex has several important special cases. |
661 |
659 |
662 \begin{thm}[Skein modules] |
660 \begin{thm}[Skein modules] |
663 \label{thm:skein-modules} |
661 \label{thm:skein-modules} |
664 \nn{linear n-categories only?} |
662 Suppose $\cC$ is a linear $n$-category |
665 The $0$-th blob homology of $X$ is the usual |
663 The $0$-th blob homology of $X$ is the usual |
666 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
664 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
667 by $\cC$. |
665 by $\cC$. |
668 \begin{equation*} |
666 \begin{equation*} |
669 H_0(\bc_*(X;\cC)) \iso A_{\cC}(X) |
667 H_0(\bc_*(X;\cC)) \iso \cl{\cC}(X) |
670 \end{equation*} |
668 \end{equation*} |
671 \end{thm} |
669 \end{thm} |
672 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. |
670 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. |
673 |
671 |
674 \begin{thm}[Hochschild homology when $X=S^1$] |
672 \begin{thm}[Hochschild homology when $X=S^1$] |