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EDIT 2

Original question below. I will award the outstanding bounty for an answer to the following question (question (2) in the OP).

Let $X$ be a Kan complex which is connected, nilpotent, and of finite $\mathbb{Q}$-type. Does there always exist a (connected, nilpotent if possible) simplicial set $Y$ with finitely many non-degenerate simplices in each simplicial dimension together with a morphism of simplicial sets $f:X\to Y$ or $f:Y\to X$ which is such that $A_\mathrm{PL}(f)$ is a quasi-isomoprhism?

If yes, a reference or an easy argument for this fact would be fantastic.

If not, are there any additional conditions that would make it true? Are there known counterexamples?

I am currently trying to relax some finiteness conditions for a result I am trying to prove in rational homotopy theory (to avoid having the result holding only for the space given by a single point) and I find myself confronted with the following two closely related questions, which I guess have already been considered many times before. A reference to an answer to (one or both of) those questions - in the positive or the negative - would be extremely welcome.

  1. Let $X$ be a simplicial set. If necessary, assume that $X$ is connected, nilpotent, and of finite $\mathbb{Q}$-type. Is there a "rational" fibrant replacement for $X$, i.e. a map $X\to Y$ that induces a quasi-isomorphism $A_{\mathrm{PL}}(Y)\to A_{\mathrm{PL}}(X)$ and such that $Y$ is Kan? it would be even better if this map were a weak equivalence of simplicial sets. Of special interest to me is the case where $X$ has finitely many non-degenerate simplices in every simplicial degree ($Y$ is of course allowed to have infinitely many).
  2. Let $X$ be a Kan complex which is connected, nilpotent, and of finite $\mathbb{Q}$-type. Is $X$ always "rationally the same" as a connected, nilpotent simplicial set of finite $\mathbb{Q}$-type with finitely many non-degenerate simplices in every simplicial dimension? More precisely, what I think I would like is the existence of a simplicial set $Y$ with finitely many non-degenerate simplices in every simplicial dimension and a morphism $f:X\to Y$ (or the other way around, or even a zig-zag of morphisms) such that $A_{\mathrm{PL}}(f)$ is a quasi-isomorphism of unital commutative algebras (resp. the same for all maps in the zig-zag). I am not asking for $f$ to be a weak equivalence (it would be very nice if it were, but I don't think it is possible in general).

Question (1) might be easy, maybe just taking any fibrant resolution or maybe taking the chains over the geometric realization of the space, but I'm not aware of the fact that $A_{PL}$ sends weak equivalences to quasi-isomorphisms (indeed, Bousfield-Gugenheim say that they suspect that it is not the case just after Prop. 8.3 in their article). I took a look in a couple of standard sources (Bousfield-Gugenheim, the book by Félix-Halperin-Thomas) but without success.

EDIT

Regarding question (1), prompted by the comment by Geoffroy Horel, we should have the following.

The functor $A_{\mathrm{PL}}$ is a right Quillen adjoint when seen as a functor $$A_{\mathrm{PL}}:sSet^{op}\longrightarrow Com\text{-}alg_{\ge0}\ ,$$ indeed, Bousfield-Gugenheim prove (Lemma 8.2 and Prop. 8.3 in their article) that its left adjoint sends cofibrations of algebras to fibrations of simplicial sets ($=$ cofibrations in the opposite category) and preserves weak equivalences, making it into a left Quillen functor.

By the Ken Brown lemma, it follows that $A_\mathrm{PL}$ preserves weak equivalences between fibrant objects in $sSet^{op}$, i.e. cofibrant objects in $sSet$. But all simplicial sets are cofibrant in the Kan model structure, so that we get that $A_{\mathrm{PL}}$ preserves weak equivalences in general.

Therefore, we can take any fibrant replacement of a simplicial set for question (1) without any issue.

Please let me know if there is any shortcoming in this argument. My doubts mainly arise from the fact that Bousfield-Gugenheim state that they suspect $A_\mathrm{PL}$ not to preserve weak equivalences, so such an easy argument looks a bit suspicious, especially considering that I think that the model categorical argument that are used were already known at the time (they should be in Quillen, but I haven't checked).

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  • $\begingroup$ I think there is a problem with the direction of the arrow in Question 1. A fibrant replacement of $X$ would be $X\to Y$. Then yes, any way of producing functorially a Kan complex replacement will work. I'm sure that $A_{PL}$ preserves weak equivalences but in that case you don't even need that as the map $X\to Y$ will be a homotopy equivalence (cofibrant source, fibrant target) and $A_{PL}$ will preserve those. $\endgroup$
    – Geoffroy Horel
    Commented Jun 22, 2020 at 10:51
  • $\begingroup$ @GeoffroyHorel Salut Geoffroy :) Thanks a lot, this makes sense! We're using the fact that it is a contravariant adjunction, right? Feel free to post it as an answer. $\endgroup$
    – Daniel Robert-Nicoud
    Commented Jun 22, 2020 at 11:05
  • $\begingroup$ Hi Daniel. Good to see you doing maths:) Yes I completely agree with what you wrote, I have no idea why Bousfield and Gugenheim got confused by this. $\endgroup$
    – Geoffroy Horel
    Commented Jun 24, 2020 at 18:57
  • $\begingroup$ @GeoffroyHorel I never really stopped ;) We're hoping to have a really nice article out this summer with Bruno, those questions are related to one of the last results. I can explain what we do to you via skype one of these days if you want, just drop me an email :) Do you know who could have an idea about question 2? $\endgroup$
    – Daniel Robert-Nicoud
    Commented Jun 24, 2020 at 22:29

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