235 in a concrete version of the homotopy colimit.) |
235 in a concrete version of the homotopy colimit.) |
236 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 |
237 yields a higher categorical and higher dimensional generalization of Deligne's |
237 yields a higher categorical and higher dimensional generalization of Deligne's |
238 conjecture on Hochschild cochains and the little 2-disks operad. |
238 conjecture on Hochschild cochains and the little 2-disks operad. |
239 |
239 |
240 \nn{needs revision} |
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241 Of course, there are currently many interesting alternative notions of $n$-category and of TQFT. |
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242 We note that our $n$-categories are both more and less general |
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243 than the ``fully dualizable" ones which play a prominent role in \cite{0905.0465}. |
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244 They are more general in that we make no duality assumptions in the top dimension $n{+}1$. |
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245 They are less general in that we impose stronger duality requirements in dimensions 0 through $n$. |
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246 Thus our $n$-categories correspond to $(n{+}\epsilon)$-dimensional {\it unoriented} or {\it oriented} TQFTs, while |
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247 Lurie's (fully dualizable) $n$-categories correspond to $(n{+}1)$-dimensional {\it framed} TQFTs. |
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248 |
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249 At several points we only sketch an argument briefly; full details can be found in \cite{1009.5025}. |
240 At several points we only sketch an argument briefly; full details can be found in \cite{1009.5025}. |
250 In this paper we attempt to give a clear view of the big picture without getting |
241 In this paper we attempt to give a clear view of the big picture without getting |
251 bogged down in technical details. |
242 bogged down in technical details. |
252 |
243 |
253 |
244 |
269 %\nn{maybe: the typical n-cat definition tries to do two things at once: (1) give a list of basic properties |
260 %\nn{maybe: the typical n-cat definition tries to do two things at once: (1) give a list of basic properties |
270 %which are weak enough to include the basic examples and strong enough to support the proofs |
261 %which are weak enough to include the basic examples and strong enough to support the proofs |
271 %of the main theorems; and (2) specify a minimal set of generators and/or axioms. |
262 %of the main theorems; and (2) specify a minimal set of generators and/or axioms. |
272 %We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second. |
263 %We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second. |
273 %More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.} |
264 %More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.} |
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265 |
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266 Of course, there are currently many interesting alternative notions of $n$-category. |
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267 We note that our $n$-categories are both more and less general |
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268 than the ``fully dualizable" ones which play a prominent role in \cite{0905.0465}. |
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269 They are more general in that we make no duality assumptions in the top dimension $n{+}1$. |
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270 They are less general in that we impose stronger duality requirements in dimensions 0 through $n$. |
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271 Thus our $n$-categories correspond to $(n{+}\epsilon)$-dimensional {\it unoriented} or {\it oriented} TQFTs, while |
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272 Lurie's (fully dualizable) $n$-categories correspond to $(n{+}1)$-dimensional {\it framed} TQFTs. |
274 |
273 |
275 We will define two variations simultaneously, as all but one of the axioms are identical in the two cases. |
274 We will define two variations simultaneously, as all but one of the axioms are identical in the two cases. |
276 These variations are ``plain $n$-categories", where homeomorphisms fixing the boundary |
275 These variations are ``plain $n$-categories", where homeomorphisms fixing the boundary |
277 act trivially on the sets associated to $n$-balls |
276 act trivially on the sets associated to $n$-balls |
278 (and these sets are usually vector spaces or more generally modules over a commutative ring) |
277 (and these sets are usually vector spaces or more generally modules over a commutative ring) |