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Suppose we have a tower of Kan fibrations between Kan complexes: $$ X_0 \xleftarrow{f_0} X_1 \xleftarrow{f_1} X_2 \xleftarrow{f_2} \dotsb $$ From this we get a commutative diagram of topological spaces: $\require{AMScd}$ \begin{CD} \left|\text{lim}_nX_n\right| @>{\alpha}>> \left|\text{holim}_nX_n\right| \\ @V{\beta}VV @VV{\gamma}V \\ \text{lim}_n|X_n| @>>{\delta}> \text{holim}_n|X_n| \end{CD} It is known that $\alpha,\beta,\gamma$ and $\delta$ are weak equivalences. What is a convenient reference for this?

Some ingredients:

  1. The geometric realization of a Kan fibration is a Serre fibration, by a short paper of Quillen with precisely that title. Thus, the spaces $|X_n|$ form a tower of Serre fibrations.
  2. The map $\alpha$ is the geometric realization of a map $\alpha_0$ of simplicial sets, and $\alpha$ is a weak equivalence iff $\alpha_0$ is a weak equivalence.
  3. There are various relevant things in the Bousfield-Kan book. In particular, Example XI.4.1(v) essentially asserts that $\alpha_0$ is a weak equivalence, but they do not really spell out a proof.
  4. If $Z$ is any one of the four spaces in the diagram, then after saying the right things about basepoints we get a short exact sequence relating $\pi_*(Z)$ to $\text{lim}$ and $\text{lim}^1$ of $\{\pi_*(X_n)\}_{n\geq 0}$. For $|\text{lim}_nX_n|$ this is Theorem IX.3.1 in Bousfield-Kan, and for $|\text{holim}_nX_n|$ it is essentially XII.7.4, although few details are given. The spaces on the bottom row can be dealt with in a similar way. Phil Hirschhorn wrote a careful and elementary argument for $\text{lim}_n|X_n|$.

This is enough to piece together a proof, but it would be nice to have a single reference where the claim is stated and proved explicitly.

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    $\begingroup$ I think there are more direct proofs: in any model category, for any tower of fibrations between fibrant objects, the comparison map between the limit and the homotopy limit is a weak equivalence; this settles $\alpha$ and $\delta$ at once (using Quillen's observation that realization of Kan fibrations are Serre fibrations). The topological realization functor being a Quillen equivalence, it commutes with homotopy limits up to weak equivalence, and this proves that $\gamma$ is a weak equivalence. This is not an answer, to your question, though. $\endgroup$
    – D.-C. Cisinski
    Commented Jul 1, 2021 at 13:52
  • $\begingroup$ @Denis-CharlesCisinski where does the proof of your first statement appear? This question arose because I thought I knew that, but then I tried to remember the proof and it did not seem so obvious. $\endgroup$
    – Neil Strickland
    Commented Jul 1, 2021 at 14:10
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    $\begingroup$ I might be mistaken, but I think it can be proven as follows. Isn't the index category of your limit a Reedy category (with only one kind of map)? Then if the maps are fibrations, then it is a Reedy fibrant diagram. Thus in that case the limit also computes the homotopy limit. $\endgroup$
    – Chris Schommer-Pries
    Commented Jul 1, 2021 at 14:23
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    $\begingroup$ In Riehl's book Categorical Homotopy Theory, the dual case (colimits vs homotopy colimits for sequences of cofibrations) is treated in Example 11.5.11 in the context of cofibrantly generated simplicial model categories. She shows in that case that it is actually a projectively cofibrant diagram. I think that in your case her argument can be adapted to show it is an injectively fibrant diagram (if the injective model structure on such diagrams exists). So again limit will compute the homotopy limit. $\endgroup$
    – Chris Schommer-Pries
    Commented Jul 1, 2021 at 14:40
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    $\begingroup$ References for what I claim above seem to be in Hirschhorn's book "Model categories and their localizations". See in particular Theorem 18.7.6 and Prop. 18.9.12. $\endgroup$
    – D.-C. Cisinski
    Commented Jul 1, 2021 at 18:30

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