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Background

Recently I asked a question on a particular construction ot the classifying space of a topological group. I got an answer, but it relied on nontrivial Quillen equivalences between various models of $(\infty,1)$-categories, and I was wondering whether there might be a more elementary approach. I then stumbled on a paper by V.Hinich (arxiv 0704.2503), which contained the line of argument I was thinking of.

Question

In the aformentioned paper, it is asserted (in Proposition 2.6.2) that for a simplicial group $G$, the $\overline{W}$-construction and the homotopy coheret nerve $\mathfrak{N}$ (applied to $G$ by regarding it as a simplicially enriched category with one object) evaluate to homotopy equivalent Kan complexes. The proof is rather short, but is unsatisfactory: The author constructs a comparison map $\overline{W}(G)\to\mathfrak{N}(G)$, computes the homotopy groups of $\overline{W}(G)$ and $\mathfrak{N}(G)$, observe that they coincide, and conclude that they have the same homotopy type without actually showing that the comparison map induces isomorphisms in the homotopy groups. Perhaps the last step might be obvious to the eyes of experienced, but it is not clear to a novice. Can someone explain why the cited proposition ought to be true?

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  • $\begingroup$ Dmitri Pavlov's answer has been turned into a paper. See arxiv.org/abs/2208.00550 $\endgroup$
    – Ken
    Commented Mar 29 at 10:20

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Both $\def\W{{\bar W}}\W$ and $\def\N{\mathfrak{N}}\N$ are right Quillen functors from the model category of simplicial groups to the model category of reduced simplicial sets (see the original paper by Dwyer–Kan, or Proposition V.6.3 in Goerss–Jardine). Thus, to show that the natural transformation $\W→\N$ is a weak equivalence, it suffices to show that the adjoint natural transformation $\def\L{{\bf L}}\L_\N→\L_\W$ of associated left adjoint functors (denoted by $\L_{(-)}$) is a natural weak equivalence.

The natural transformation $\L_\N→\L_\W$ is a natural transformation of left Quillen functors from simplicial sets to simplicial groups. Since reduced simplicial sets are generated by $\def\Z{{\bf Z}}\W\Z≃S^1$ under homotopy colimits, to show that $\L_\N→\L_\W$ is a natural weak equivalence, it suffices to show that $\L_\N(S^1)→\L_\W(S^1)$ is a weak equivalence of simplicial sets. This is done by a simple direct inspection.

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  • $\begingroup$ Thanks again, Dmitri! I am afraid it is not the case that $\overline{W}$ is right Quillen when viewed as a functor $\mathsf{sGrp}\to\mathsf{sSet}$. (See Goerss-Jardine, Chapter V, Corollary 6.9.) $\endgroup$
    – Ken
    Commented Jul 7, 2022 at 6:08
  • $\begingroup$ @Ken: Forgot to say “reduced”, which is necessary here. Corrected now and added references. $\endgroup$
    – Dmitri Pavlov
    Commented Jul 7, 2022 at 6:28
  • $\begingroup$ May I ask you why the category of reduced simplicial sets is generated under homotopy colimits by $\Delta^0$? $\endgroup$
    – Ken
    Commented Jul 7, 2022 at 6:51
  • $\begingroup$ @Ken: Generated by S^1 (forgot to adjust this part once I adjusted the reduced part). Sets of generators can be transported along Quillen equivalences such as $\bar W$, and Z generates simplicial groups under homotopy colimits. $\endgroup$
    – Dmitri Pavlov
    Commented Jul 7, 2022 at 13:36
  • $\begingroup$ Wow, this is utterly amazing! Thank you! $\endgroup$
    – Ken
    Commented Jul 8, 2022 at 2:29

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