219 We then describe how to use [homotopy] colimits to extend $n$-categories |
219 We then describe how to use [homotopy] colimits to extend $n$-categories |
220 from balls to arbitrary $k$-manifolds. |
220 from balls to arbitrary $k$-manifolds. |
221 This extension is the desired derived version of a TQFT, which we call the blob complex. |
221 This extension is the desired derived version of a TQFT, which we call the blob complex. |
222 (The name comes from the ``blobs" which feature prominently |
222 (The name comes from the ``blobs" which feature prominently |
223 in a concrete version of the homotopy colimit.) |
223 in a concrete version of the homotopy colimit.) |
224 |
224 We then review some basic properties of the blob complex, and finish by showing how it |
225 \nn{In many places we omit details; they can be found in MW. |
225 yields a higher categorical and higher dimensional generalization of Deligne's |
226 (Blanket statement in order to avoid too many citations to MW.)} |
226 conjecture on Hochschild cochains and the little 2-disks operad. |
227 |
227 |
228 \nn{perhaps say something explicit about the relationship of this paper to big blob paper. |
228 \nn{maybe this is not necessary?} |
229 like: in this paper we try to give a clear view of the big picture without getting bogged down in details} |
229 In an attempt to forestall any confusion that might arise from different definitions of |
230 |
230 ``$n$-category" and ``TQFT", we note that our $n$-categories are both more and less general |
231 \nn{diff w/ lurie} |
231 than the ``fully dualizable" ones which play a prominent role in \cite{0905.0465}. |
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232 More general in that we make no duality assumptions in the top dimension $n+1$. |
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233 Less general in that we impose stronger duality requirements in dimensions 0 through $n$. |
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234 Thus our $n$-categories correspond to $(n{+}\epsilon)$-dimensional unoriented or oriented TQFTs, while |
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235 Lurie's (fully dualizable) $n$-categories correspond to $(n{+}1)$-dimensional framed TQFTs. |
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236 |
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237 Details missing from this paper can usually be found in \cite{1009.5025}. |
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238 |
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239 %\nn{In many places we omit details; they can be found in MW. |
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240 %(Blanket statement in order to avoid too many citations to MW.)} |
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241 % |
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242 %\nn{perhaps say something explicit about the relationship of this paper to big blob paper. |
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243 %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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244 |
232 |
245 |
233 \section{Definitions} |
246 \section{Definitions} |
234 \subsection{$n$-categories} \mbox{} |
247 \subsection{$n$-categories} \mbox{} |
235 |
248 |
236 \nn{rough draft of n-cat stuff...} |
249 \nn{rough draft of n-cat stuff...} |