239 properties of identity morphisms to low dimensions, we need regular products, |
239 properties of identity morphisms to low dimensions, we need regular products, |
240 pinched products and even half-pinched products. |
240 pinched products and even half-pinched products. |
241 I'm not sure what the best way to cleanly axiomatize the properties of these various is. |
241 I'm not sure what the best way to cleanly axiomatize the properties of these various is. |
242 For the moment, I'll assume that all flavors of the product are at |
242 For the moment, I'll assume that all flavors of the product are at |
243 our disposal, and I'll plan on revising the axioms later.} |
243 our disposal, and I'll plan on revising the axioms later.} |
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244 |
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245 \nn{current idea for fixing this: make the above axiom a ``preliminary version" |
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246 (as we have already done with some of the other axioms), then state the official |
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247 axiom for maps $\pi: E \to X$ which are almost fiber bundles. |
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248 one option is to restrict E to be a (full/half/not)-pinched product (up to homeo). |
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249 the alternative is to give some sort of local criterion for what's allowed. |
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250 state a gluing axiom for decomps $E = E'\cup E''$ where all three are of the correct type. |
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251 } |
244 |
252 |
245 All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |
253 All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |
246 The last axiom (below), concerning actions of |
254 The last axiom (below), concerning actions of |
247 homeomorphisms in the top dimension $n$, distinguishes the two cases. |
255 homeomorphisms in the top dimension $n$, distinguishes the two cases. |
248 |
256 |