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There are three well known model structures in the category of spaces - the Quillen's structure, the Strom's structure and the mixed structure. I was wondering if there is some other nice structures. In particular, for a fixed $n\in\mathbb{N} $, is there a model structure in Top such that the weak equivalences are n-equivalences and the cofibrations are closed Hurewicz cofibrantions (or other nice cofibration)?

Obs.: By n-equivalence, I mean a continous function that induces isomorphisms between t-homotopy groups ($ t\leq n-1 $) and induces a surjective morphism between n-homotopy groups.

Is there any text about other useful model structures in Top?

Thank you

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    $\begingroup$ If there is such a model structure it can't be left proper since homotopy excision would force you to lose one level of your equivalence (in general the cobase-changed morphism would be an (n-1)-equivalence instead of an n-equivalence.) $\endgroup$
    – Dylan Wilson
    Commented Apr 1, 2013 at 18:15
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    $\begingroup$ Also Theorem 1.2 of this paper seems relevant: arxiv.org/pdf/1303.1286.pdf $\endgroup$
    – Dylan Wilson
    Commented Apr 1, 2013 at 18:35
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    $\begingroup$ In the simplicial setting, Quillen showed the existence of such model structures in his seminal long paper on rational homotopy theory. In the topoligical setting you have Extremiana Aldana, J. Ignacio; Hernández Paricio, L. Javier; Rivas Rodríguez, M. Teresa Closed model categories for [n,m]-types. Theory Appl. Categ. 3 (1997), No. 10, 250–268. $\endgroup$
    – Fernando Muro
    Commented Apr 1, 2013 at 18:43
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    $\begingroup$ However, the topological paper cited gives a fairly nice description of fibrations but not such a nice one for cofibrations. I haven't looked at Quillen's. $\endgroup$
    – Dylan Wilson
    Commented Apr 1, 2013 at 18:49
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    $\begingroup$ The class of $n$-equivalences, as you defined them, does not satisfy the 2-out-of-3 property, so that there is no model structure for which your $n$-equivalences are the weak equivalences. If you take the closure of $n$-equivalences in your sense by the 2-out-of-3 property, then you recover what I would rather call the $(n−1)$-equivalences: continuous maps which induce an isomorphism on homotopy groups in degree $<n$. The latter class corresponds to the left Bousfield localisation of the Quillen model category structure by the map $S^n\to pt$. $\endgroup$
    – D.-C. Cisinski
    Commented Apr 1, 2013 at 19:25

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A different way of getting a model structure which captures "truncation" data was asked about here, and I answered it. There a $\pi_n$ weak equivalence is an isomorphism for $t\leq n$ but we make no mention of surjection in top degree (as Denis Charles Cisinski points out in his comment, this is the right notion of weak equivalence for your truncated model structure). The content of my answer was figuring out what a $\pi_n$ fibration should be to make it a model structure. We also discussed "going to the limit" and recovering the usual Quillen model structure. And we discussed the other kind of truncations, where you have a $\pi_t$ isomorphism for all $t>n$. There are a number of references on that answer which should help you.

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    $\begingroup$ Fernando Muro's comment is interesting. The paper referenced in my answer on that other MO thread was by some of the same authors but in 1995 rather than 1997. I guess the paper Fernando mentions will have a clearer view of things, since it came out later. $\endgroup$
    – David White
    Commented Apr 1, 2013 at 20:25

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