Timeline for Simplicial model categories and simplicial equivalence
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16 events
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| Sep 12, 2018 at 8:28 | comment | added | S. Douteau | @DmitriPavlov You are right, I should not use the term homotopy equivalences for these maps. However, what I meant by "closure by transitivity" is that when one wants to compute actual sets of homotopy classes of maps between two spaces, one needs to quotient out the set of maps by the homotopy relation. And to do so, a closure by transitivity is needed unless both spaces are bifibrant. Furthermore, this set of homotopy classes is exactly $\pi_0Map_M(a,b)$, and even though it does not correspond to the Homset in the homotopy category, there is still a well defined map $[X,Y]\to Hom_{Ho}(X,Y)$ | |
| Sep 11, 2018 at 18:01 | vote | accept | Let | ||
| Sep 11, 2018 at 17:25 | answer | added | Let | timeline score: 2 | |
| Sep 11, 2018 at 14:25 | comment | added | Dmitri Pavlov | @S.Douteau: I am using the standard definition of simplicial homotopy equivalence: a map f:X→Y such that there is g:Y→X and h_X:Δ^1×X→X and h_Y:Δ^1×Y→Y that are homotopies between id_X and gf respectively fg and id_Y. This yields the correct notion for Kan complexes. "Closure by transitivity" of homotopy equivalences for non-Kan complexes does not give the correct notion: there are many pairs of maps f,g:X→Y that become identical maps in Ho(sSet), but are not homotopic in your sense (e.g., one could enhance my example by taking infinite disjoint unions of simplices of increasing dimension). | |
| Sep 11, 2018 at 10:10 | comment | added | S. Douteau | @DmitriPavlov: I am not sure what definition of simplicial homotopy equivalence you are using, but for the sake of computing $\pi_0$, which relies on taking a closure by transitivity of the notion of homotopy equivalence, the inclusion of $\Delta^0$ into $\Delta^n$ is an homotopy equivalence. You can always write it as a composition of "elementary" homotopy equivalence by writing $\Delta^0\to \Delta^1\to \dots\to \Delta^n$. | |
| Sep 9, 2018 at 15:33 | comment | added | Dmitri Pavlov | @Amadeus: π_0(sSet) contains Ho(sSet) as a full subcategory, but is much bigger than Ho(sSet), essentially because nonfibrant simplicial sets are poorly behaved with respect to simplicial homotopies. For instance, the inclusion of Δ^0 into Δ^n as an interior vertex i (0<i<n) is not a simplicial homotopy equivalence. Kan's Ex^∞ functor is a simplicial fibrant replacement functor for simplicial sets. | |
| Sep 9, 2018 at 15:05 | comment | added | Let | @DmitriPavlov I'm learning model categories, I found out that I should start with simplicial model categories since I have some problems to digest the technical definitions of a general model category. My question is maybe naive for the experts. I wanted just to see what is the difference between $\pi_{0}M$ and $Ho(M)$ and how much information is missed passing from $\pi_{0}M$ to the full subcategory $Ho(M)$. I have edited my question with a potential counterexample. | |
| Sep 9, 2018 at 14:53 | comment | added | Dmitri Pavlov | It would be helpful if some context was provided as to the origin of this question. Why would we want to consider π_0(M) in the first place? This is a fairly atypical thing to do to a simplicial model category. | |
| Sep 9, 2018 at 12:37 | history | edited | Let | CC BY-SA 4.0 |
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| Sep 8, 2018 at 15:38 | history | edited | Let | CC BY-SA 4.0 |
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| Sep 8, 2018 at 15:24 | comment | added | Niall Taggart | I think the clarification for $Map$ comes from different authors using it for different things, for example, simplicial enrichment, homotopy function complexes, etc etc | |
| Sep 8, 2018 at 15:05 | comment | added | Let | @DenisNardin M is a simplicial model category, $Map$ is given enrichment by definition. Do I misunderstand your question ? | |
| Sep 8, 2018 at 15:01 | comment | added | Denis Nardin | How are you defining $\mathrm{Map}$? In general you'll need to take some kind of (co)fibrant replacement of the arguments to get the correct homotopy type of the mapping space... | |
| Sep 8, 2018 at 13:01 | history | edited | Let | CC BY-SA 4.0 |
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| Sep 8, 2018 at 12:45 | history | edited | Let | CC BY-SA 4.0 |
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| Sep 8, 2018 at 12:00 | history | asked | Let | CC BY-SA 4.0 |