Timeline for Exit path categories of regular CW complexes
Current License: CC BY-SA 4.0
7 events
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| Jan 3, 2023 at 14:37 | comment | added | Markus Zetto | Concerning my last comment: I think that after we realize that $X \simeq |\operatorname{Fac}[X]| \simeq |\operatorname{Ent}[X]|$, we can deduce the desired result by combining 4.22 in arxiv.org/abs/2112.02394 to see that $\operatorname{Sing}^S(X)$, $\operatorname{Fac[X]}$ and $\operatorname{Ent[X]}$ are Joyal-Kan equivalent, with 2.5.4 arxiv.org/abs/1811.01119. | |
| Jan 3, 2023 at 13:19 | comment | added | Markus Zetto | That does look very helpful, thank you! Stratified homotopy equivalence of the realizations as far as I know only shows that Ent[X] and Fac[X] are Joyal-Kan-equivalent to the exit-path category, not Joyal equivalent - but I think this last step might be straightforward using a fibrant replacement argument since the fibers are contractible in both cases. I am just trying to write down a different proof in Lurie's convention, but I think one could solve it that way. | |
| Jan 3, 2023 at 13:02 | answer | added | Markus Zetto | timeline score: 1 | |
| Jan 3, 2023 at 12:33 | comment | added | Vidit Nanda | If you're willing to use entrance paths instead of exit paths, Sec 3.1 of this paper gives a precise definition of the combinatorial version of the entrance path category of a regular CW complex: arxiv.org/abs/1510.01907. For my purposes it was enough to use a poset-enriched category, but it is straightforward to turn this into an $\infty$-category if you want. The fact that the classifying space of this category is homotopy equivalent to the original complex is Prop 3,3. I'm happy to say more in an answer if this is what you're looking for! | |
| Jan 3, 2023 at 12:19 | comment | added | Markus Zetto | The reference in the mentioned paper was for DAG, not HA - this resolves the side question (I made an edit to indicate this) and might be helpful for the actual problem. | |
| Jan 3, 2023 at 12:17 | history | edited | Markus Zetto | CC BY-SA 4.0 |
corrected a reference which resolves one question
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| Jan 2, 2023 at 13:08 | history | asked | Markus Zetto | CC BY-SA 4.0 |