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251 we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
251 we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
252 instead of $C_*(\Homeo_\bd(X))$. |
252 instead of $C_*(\Homeo_\bd(X))$. |
253 Taking singular chains converts a space-type $A_\infty$ $n$-category into a chain complex |
253 Taking singular chains converts a space-type $A_\infty$ $n$-category into a chain complex |
254 type $A_\infty$ $n$-category. |
254 type $A_\infty$ $n$-category. |
255 |
255 |
256 |
256 \medskip |
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257 |
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258 The alert reader will have already noticed that our definition of (plain) $n$-category |
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259 is extremely similar to our definition of topological fields. |
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260 The only difference is that for the $n$-category definition we restrict our attention to balls |
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261 (and their boundaries), while for fields we consider all manifolds. |
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262 \nn{also: difference at the top dimension; fix this} |
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263 Thus a system of fields determines an $n$-category simply by restricting our attention to |
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264 balls. |
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265 The $n$-category can be thought of as the local part of the fields. |
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266 Conversely, given an $n$-category we can construct a system of fields via |
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267 \nn{gluing, or a universal construction} |
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268 |
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269 \nn{Next, say something about $A_\infty$ $n$-categories and ``homological" systems |
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270 of fields. |
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271 The universal (colimit) construction becomes our generalized definition of blob homology. |
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272 Need to explain how it relates to the old definition.} |
257 |
273 |
258 \medskip |
274 \medskip |
259 |
275 |
260 \hrule |
276 \hrule |
261 |
277 |
273 \item blob complex is an example of an $A_\infty$ $n$-category |
289 \item blob complex is an example of an $A_\infty$ $n$-category |
274 \item fundamental $n$-groupoid is an example of an $A_\infty$ $n$-category |
290 \item fundamental $n$-groupoid is an example of an $A_\infty$ $n$-category |
275 \item traditional $n$-cat defs (e.g. *-1-cat, pivotal 2-cat) imply our def of plain $n$-cat |
291 \item traditional $n$-cat defs (e.g. *-1-cat, pivotal 2-cat) imply our def of plain $n$-cat |
276 \item conversely, our def implies other defs |
292 \item conversely, our def implies other defs |
277 \item traditional $A_\infty$ 1-cat def implies our def |
293 \item traditional $A_\infty$ 1-cat def implies our def |
278 \item ... and vice-versa |
294 \item ... and vice-versa (already done in appendix) |
279 \item say something about unoriented vs oriented vs spin vs pin for $n=1$ (and $n=2$?) |
295 \item say something about unoriented vs oriented vs spin vs pin for $n=1$ (and $n=2$?) |
280 \item spell out what difference (if any) Top vs PL vs Smooth makes |
296 \item spell out what difference (if any) Top vs PL vs Smooth makes |
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297 \item explain relation between old-fashioned blob homology and new-fangled blob homology |
281 \end{itemize} |
298 \end{itemize} |
282 |
299 |
283 |
300 |