244 |
244 |
245 |
245 |
246 \section{Definitions} |
246 \section{Definitions} |
247 \subsection{$n$-categories} \mbox{} |
247 \subsection{$n$-categories} \mbox{} |
248 |
248 |
249 \nn{rough draft of n-cat stuff...} |
249 In this section we give a definition of $n$-categories designed to work well with TQFTs. |
250 |
250 The main idea is to base the definition on actual balls, rather combinatorial models of them. |
251 \nn{maybe say something about goals: well-suited to TQFTs; avoid proliferation of coherency axioms; |
251 This has the advantages of avoiding a proliferation of coherency axioms and building in a strong |
252 non-recursive (n-cats not defined n terms of (n-1)-cats; easy to show that the motivating |
252 version of duality from the start. |
253 examples satisfy the axioms; strong duality; both plain and infty case; |
253 |
254 (?) easy to see that axioms are correct, in the sense of nothing missing (need |
254 |
255 to say this better if we keep it)} |
255 %\nn{maybe say something about goals: well-suited to TQFTs; avoid proliferation of coherency axioms; |
256 |
256 %non-recursive (n-cats not defined n terms of (n-1)-cats; easy to show that the motivating |
257 \nn{maybe: the typical n-cat definition tries to do two things at once: (1) give a list of basic properties |
257 %examples satisfy the axioms; strong duality; both plain and infty case; |
258 which are weak enough to include the basic examples and strong enough to support the proofs |
258 %(?) easy to see that axioms are correct, in the sense of nothing missing (need |
259 of the main theorems; and (2) specify a minimal set of generators and/or axioms. |
259 %to say this better if we keep it)} |
260 We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second. |
260 % |
261 More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.} |
261 %\nn{maybe: the typical n-cat definition tries to do two things at once: (1) give a list of basic properties |
262 |
262 %which are weak enough to include the basic examples and strong enough to support the proofs |
263 \nn{say something about defining plain and infty cases simultaneously} |
263 %of the main theorems; and (2) specify a minimal set of generators and/or axioms. |
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264 %We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second. |
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265 %More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.} |
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266 |
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267 We will define plain and $A_\infty$ $n$-categories simultaneously, as all but one of the axioms are identical |
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268 in the two cases. |
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269 |
264 |
270 |
265 There are five basic ingredients |
271 There are five basic ingredients |
266 \cite{life-of-brian} of an $n$-category definition: |
272 \cite{life-of-brian} of an $n$-category definition: |
267 $k$-morphisms (for $0\le k \le n$), domain and range, composition, |
273 $k$-morphisms (for $0\le k \le n$), domain and range, composition, |
268 identity morphisms, and special behavior in dimension $n$ (e.g. enrichment |
274 identity morphisms, and special behavior in dimension $n$ (e.g. enrichment |