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Consider in the category sSet of simplicial sets, let $X$ be a $J$-indexed diagram in sSet ($J$ a small category), and for each $j\in J$, $X_j$ is a Kan complex, then do we have

$\pi_0({\rm holim}_{j\in J} X_j)\xrightarrow{\cong}{\rm lim}_{j\in J} \pi_0(X_j)$?

I can show this for $J$ being the poset $\mathbb{N}^{\rm op}$ as one can replace the diagram in such a way that all the maps in the (replaced) diagram are fibration, due to its simple shape.

Is it true in general?

(For a Kan complex $K$, $\pi_0K$ is the (reflexive) coequalizer of the two faces $K_1\to K_0$. Reflexive coequalizers in Set commute with finite products.)

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    $\begingroup$ It's not even true for your poset: look at the Kan complexes associated to chain complexes, and then $\lim^1$ gives an obstruction. For a simpler counterexample, contemplate the diagram $* \to BG \leftarrow *$. $\endgroup$ Oct 15, 2019 at 20:09
  • $\begingroup$ Your simple counterexample is very nice, thanks! But I think my poset case is correct, we can use functorial factorization to get a fibrant object in the "tower" category, whose limit is (weakly equivalent to) the homotopy limit of the given sequence. And for a tower of fibration of fibrant simplicial sets (Kan complex), one can show the result using that a fibration is surjection on components (meaning that on each degree, a component of the source is mapped surjectively to a component of the target). $\endgroup$
    – Lao-tzu
    Oct 15, 2019 at 20:38
  • $\begingroup$ A fibration isn't necessarily a surjection on components. $\endgroup$ Oct 15, 2019 at 20:46
  • $\begingroup$ I mean the image of a Kan fibration is a disjoint union of some components of the target. (Of course, the map from empty simplicial set to any simplicial set is a Kan fibration.) $\endgroup$
    – Lao-tzu
    Oct 15, 2019 at 20:50
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    $\begingroup$ But you can't guarantee injectivity of $\pi_0(ho\lim) \to \lim \pi_0$, as you can get a contribution from $\lim^1\pi_1$. $\endgroup$ Oct 15, 2019 at 21:16

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