Timeline for Homotopy theory of non-test categories?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Apr 13, 2017 at 12:32 | comment | added | Artur Jackson | @TimCampion, I agree that this is not so enlightening. I think one should find better examples to get a better feel for the issue. | |
| Apr 5, 2017 at 6:36 | comment | added | Tim Campion | This is interesting, but I'm not sure how enlightening it is. As Andreas points out, in order to get this result, you need to change the basic localizer on $\mathsf{Cat}$ you're using. In fact, you need to change it to the localizer $\mathcal{W}_n$ which presents $n$-types on $\mathsf{Cat}$. With this localizer, I presume that the induced Cisinski model structure for the test category $\Delta$ also presents $n$-types, so it's not surprising that the model structure for the local test category $\Delta_{\leq n+1}$ does too. | |
| Apr 5, 2017 at 6:29 | history | edited | Artur Jackson | CC BY-SA 3.0 |
Added reference.
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| Apr 5, 2017 at 6:15 | comment | added | Artur Jackson | Andrea, am I off by one here? | |
| Apr 4, 2017 at 17:06 | comment | added | Andrea Gagna | I should add that the class of functors you pullback to presheaves categories in this case is not the class of Thomason weak equivalences described in the OP, but instead a further (left Boudfield) localisation. | |
| Apr 4, 2017 at 16:56 | comment | added | Andrea Gagna | It is instead $\Delta_{\leqslant n+1}$. Indeed, both the pairs $(i^*, i_*)$ and $(i_!, i^*)$ are Quillen equivalences for the model structures of $n$-equivalences, where here $i\colon \Delta_{\leqslant n+1} \to \Delta$ is the canonical embedding. This is proven, for instance, in Corollary 9.2.7 of Cisinski's book Les préfaisceaux comme modèles des types d'homotopie . | |
| Apr 1, 2017 at 15:31 | comment | added | Marc Hoyois | I don't think I believe this. In general there is a conservative functor (of ∞-categories) $\mathcal C[\mathcal W^{-1}] \to Spaces$ whose image consists of all spaces that are homotopy colimits of $C^{op}$-diagrams of sets. When $C=\Delta_{\leq n}$, these are the spaces of homotopy dimension $\leq n$, and it seems unlikely that there is a conservative functor from $n$-types to spaces of homotopy dimension $\leq n$. What is your reasoning? | |
| Apr 1, 2017 at 5:07 | comment | added | Artur Jackson | I'm curious if this fits with Anton's comments! | |
| Apr 1, 2017 at 5:06 | history | answered | Artur Jackson | CC BY-SA 3.0 |