Timeline for Homotopy theory of non-test categories?
Current License: CC BY-SA 3.0
14 events
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| Dec 11, 2021 at 14:53 | comment | added | D.-C. Cisinski | Andrea's comments above are not completely accurate: in the case of a local test category, we get a model structure which models spaces over the nerve of $\mathcal C$. For information, the nlab page on test categories now contains a summary of the main results in English. As for presheaves on finite sets, we also model structures with all monomorphisms as cofibrations, but the version with fewer cofibrations is nicer because we have explicit generators for trivial cofibrations. | |
| Apr 5, 2017 at 6:47 | comment | added | Tim Campion | Cisinski gives a model structure on presheaves on $\Gamma^\mathrm{op} = \mathsf{FinSet}$ which is Quillen-equivalent to spaces... only certain monomorphisms are cofibrations; the construction uses his notion of generalized Reedy category rather than his general presheaf machinery. But that's on the opposite category, and on $\Gamma$ itself of course one is going for connective spectra... | |
| Apr 5, 2017 at 6:40 | comment | added | Artur Jackson | @TimCapion: I'm quite interested in this question for a number of reasons. Lars Hesselholt asked me once if (very) special $\Gamma$-spaces/sets can be seen as the sheaves on $\Gamma = \mathbf{Fin}^{op}$ for some topology. I don't think this $\Gamma$ is a test category. (Although, I haven't checked.) I'm curious to know what Cisinski's associated homotopy theory looks like! May or may not be useful: Some people think that the inclusion of $\Gamma$-sets into $\Gamma$-spaces is essentially surjective on homotopy classes. | |
| Apr 5, 2017 at 6:25 | comment | added | Artur Jackson | I think that is essentially the shape. I think I learned this from a note by Marc Hoyois :). "A note on étale homotopy." I think this was subsumed into Higher Galois Theory? @MarcHoyois ? One needs to be a bit careful though, I believe, when tracing through the classical constructions of Artin-Mazur, Friedlander, etc. Artin-Mazur first produced pro-homotopy types. Then Friedlander "rigidified" this to give a (pro-)topological type. Fancier constructions upgrade this construction to an $\infty$-functor. And the former is the so called `shape.' | |
| Apr 4, 2017 at 16:44 | comment | added | Andrea Gagna | What Cisinski proves in his book Les préfaisceaux comme modèles des types d'homotopie is that the category of presheaves of a small category $A$ is equipped with a model structure such that cofibrations are monomorphisms and the weak equivalences are the morphisms you describe if and only if $A$ is local test (and it always model homotopy types). It is not true in general: think about the category of semi-simplices $\Delta'$, i.e. the subcategory of simplices where you consider only non-decreasing maps. The category of semi-simplicial presheaves do not allow a model structure of that kind. | |
| Apr 4, 2017 at 1:34 | comment | added | Tim Campion | @ArturJackson That's what I mean -- or perhaps a refinement along the lines of Bousfield-Friedlander. My terminology is probably nonstandard. I'm pretty sure this is the same as the shape of the topos regarded as a locally discrete $\infty$-topos. | |
| Apr 1, 2017 at 18:08 | history | edited | David White | CC BY-SA 3.0 |
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| Apr 1, 2017 at 5:06 | answer | added | Artur Jackson | timeline score: 3 | |
| Apr 1, 2017 at 5:03 | comment | added | Artur Jackson | What is the étale homotopy type of a topos? Is this the Artin-Mazur pro-homotopy type? | |
| Mar 31, 2017 at 19:10 | comment | added | Anton Fetisov | In fact, if you just consider presheaves of sets on a discrete category $C$, then the described construction is tautological: you get again $[C^{op}, Set]$, which is purely discrete and 1-categorical. I'm not sure how one can formulate a theorem for a general $C$ that would nicely interpolate between a purely homotopy-theoretic and a purely discrete world. | |
| Mar 31, 2017 at 19:08 | comment | added | Anton Fetisov | If I'd take a guess I'd say that for moderately good $C$'s the corresponding theory would be the subcategory of spaces generated by spaces of the form $P\otimes N(C/\cdot)$, where $N$ is the nerve, $P: [C^{op}, Set]$, $\otimes$ means the homotopy coend (which can be described just as a homotopy colimit over a different category) and $C/\cdot$ is the covariant functor of overcategory. The problems is that 1). I see no reason why all maps between spaces would be generated this way, in fact often they won't; 2). objects of $C$ may have equivalent nerves, but not connected by a chain of equivs. | |
| Mar 31, 2017 at 19:02 | comment | added | Tim Campion | Agh, of course! For now, let's just assume we're talking about presheaf toposes so we have a canoncial site $C$ for the topos $\mathcal{C} = [C^\mathrm{op},\mathsf{Set}]$, and we can take $i_\mathcal{C}(X) = C/X$ as you say. I think Cisinski has some work that's not restricted just to presheaf toposes, though. I will track it down and edit my question accordingly. | |
| Mar 31, 2017 at 18:35 | comment | added | Anton Fetisov | Since $C/X$ always has a final object $X \xrightarrow{1} X$, its nerve is contractible. As described, all objects of $C$ will be weakly equivalent. In the definition of a test category one considers presheaves $[C^{op}, Set]$ and overcategories only of the form $C/X$. I assume you're asking what is the homotopy theory of presheaves on $C$ for a general $C$ via the above construction. | |
| Mar 31, 2017 at 17:51 | history | asked | Tim Campion | CC BY-SA 3.0 |