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275 Put another way, the cell decompositions we consider are dual to standard cell |
275 Put another way, the cell decompositions we consider are dual to standard cell |
276 decompositions of $X$. |
276 decompositions of $X$. |
277 |
277 |
278 We will always assume that our $n$-categories have linear $n$-morphisms. |
278 We will always assume that our $n$-categories have linear $n$-morphisms. |
279 |
279 |
280 \nn{need to replace ``cell decomposition" below with something looser. not sure what to call it. |
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281 maybe ``nice stratification"?? the link of each piece of each stratum should be a cell decomposition of |
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282 a sphere, but that's probably all we need. or maybe refineable to a cell decomp?} |
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283 |
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284 For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with |
280 For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with |
285 an object (0-morphism) of the 1-category $C$. |
281 an object (0-morphism) of the 1-category $C$. |
286 A field on a 1-manifold $S$ consists of |
282 A field on a 1-manifold $S$ consists of |
287 \begin{itemize} |
283 \begin{itemize} |
288 \item a cell decomposition of $S$ (equivalently, a finite collection |
284 \item a cell decomposition of $S$ (equivalently, a finite collection |
354 \item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with |
350 \item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with |
355 domain and range determined by the labelings of the link of $j$-cell. |
351 domain and range determined by the labelings of the link of $j$-cell. |
356 \end{itemize} |
352 \end{itemize} |
357 |
353 |
358 |
354 |
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355 It is customary when drawing string diagrams to omit identity morphisms. |
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356 In the above context, this corresponds to erasing cells which are labeled by identity morphisms. |
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357 The resulting structure might not, strictly speaking, be a cell complex. |
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358 So when we write ``cell complex" above we really mean a stratification which can be |
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359 refined to a genuine cell complex. |
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360 |
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361 |
359 |
362 |
360 \subsection{Local relations} |
363 \subsection{Local relations} |
361 \label{sec:local-relations} |
364 \label{sec:local-relations} |
362 |
365 |
363 For convenience we assume that fields are enriched over Vect. |
366 For convenience we assume that fields are enriched over Vect. |