213 |
213 |
214 This paper is organized as follows. |
214 This paper is organized as follows. |
215 We first give an account of our version of $n$-categories. |
215 We first give an account of our version of $n$-categories. |
216 According to our definition, $n$-categories are, among other things, |
216 According to our definition, $n$-categories are, among other things, |
217 functorial invariants of $k$-balls, $0\le k \le n$, which behave well with respect to gluing. |
217 functorial invariants of $k$-balls, $0\le k \le n$, which behave well with respect to gluing. |
218 We then describe how to use [homotopy] colimits to extend $n$-categories |
218 We then show how to extend an $n$-category from balls to arbitrary $k$-manifolds, |
219 from balls to arbitrary $k$-manifolds. |
219 using colimits and homotopy colimits. |
220 This extension is the desired derived version of a TQFT, which we call the blob complex. |
220 This extension, which we call the blob complex, has as $0$-th homology the usual TQFT invariant. |
221 (The name comes from the ``blobs" which feature prominently |
221 (The name comes from the ``blobs" which feature prominently |
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?} |
227 \nn{maybe this is not necessary?} \nn{let's move this to somewhere later, if we keep it} |
228 In an attempt to forestall any confusion that might arise from different definitions of |
228 In an attempt to forestall any confusion that might arise from different definitions of |
229 ``$n$-category" and ``TQFT", 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}. |
230 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 $n+1$. |
231 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 $n$. |
232 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 |
233 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. |
234 Lurie's (fully dualizable) $n$-categories correspond to $(n{+}1)$-dimensional framed TQFTs. |
235 |
235 |
236 Details missing from this paper can usually be found in \cite{1009.5025}. |
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. |
237 |
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238 %\nn{In many places we omit details; they can be found in MW. |
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239 %(Blanket statement in order to avoid too many citations to MW.)} |
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240 % |
|
241 %\nn{perhaps say something explicit about the relationship of this paper to big blob paper. |
|
242 %like: in this paper we try to give a clear view of the big picture without getting bogged down in details} |
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243 |
237 |
244 |
238 |
245 \section{Definitions} |
239 \section{Definitions} |
246 \subsection{$n$-categories} \mbox{} |
240 \subsection{$n$-categories} \mbox{} |
247 |
241 |
288 Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with |
282 Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with |
289 a product of $k$ intervals (c.f. \cite{0909.2212}) but rather with any $k$-ball, that is, any $k$-manifold which is homeomorphic |
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 |
290 to the standard $k$-ball $B^k$. |
284 to the standard $k$-ball $B^k$. |
291 \nn{maybe add that in addition we want functoriality} |
285 \nn{maybe add that in addition we want functoriality} |
292 |
286 |
293 We haven't said precisely what sort of balls we are considering, |
287 By default our balls are oriented, |
294 because we prefer to let this detail be a parameter in the definition. |
288 but it is useful at times to vary this, |
295 It is useful to consider unoriented, oriented, Spin and $\mbox{Pin}_\pm$ balls. |
289 for example by considering unoriented or Spin balls. |
296 Also useful are more exotic structures, such as balls equipped with a map to some target space, |
290 We can also consider more exotic structures, such as balls with a map to some target space, |
297 or equipped with $m$ independent vector fields. |
291 or equipped with $m$ independent vector fields. |
298 (The latter structure would model $n$-categories with less duality than we usually assume.) |
292 (The latter structure would model $n$-categories with less duality than we usually assume.) |
299 |
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300 %In fact, the axioms here may easily be varied by considering balls with structure (e.g. $m$ independent vector fields, a map to some target space, etc.). Such variations are useful for axiomatizing categories with less duality, and also as technical tools in proofs. |
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301 |
293 |
302 \begin{axiom}[Morphisms] |
294 \begin{axiom}[Morphisms] |
303 \label{axiom:morphisms} |
295 \label{axiom:morphisms} |
304 For each $0 \le k \le n$, we have a functor $\cC_k$ from |
296 For each $0 \le k \le n$, we have a functor $\cC_k$ from |
305 the category of $k$-balls and |
297 the category of $k$-balls and |