Consider a small diagram $X_\bullet$ of simplicial sets. Under what sufficient conditions (apart from $X_\bullet$ being injective fibrant), is the map $\text{lim }X_\bullet \to \text{holim } X_\bullet$ a weak equivalence? Is it sufficient that the morphisms of simplicial sets $X_{i} \rightarrow X_{j}$ be fibrations $\forall i,j$?

I'm looking for something analogous to what one has in case of homotopy pullbacks of simplicial sets where one map being a fibration is sufficient to ensure that homotopy pullback is same as categorical pullback.

EDIT: I'm interested in diagrams indexed by cosimplex category $\Delta$.

  • 1
    $\begingroup$ With your edit, what you want look at are Reedy fibr ant objects. For a reedy indexing category the Reedy model structure can be used to compute the holim, hence taking a fibrant replacement for this model structure is enough. And being fibrant in the Reedy model structure is a quite explicit condition. $\endgroup$ – Simon Henry May 16 '18 at 5:00
  • $\begingroup$ @Simon Henry, but I'm looking for something weaker than diagram being Reedy fibrant, because the diagram that i'm working with is not Reedy fibrant. $\endgroup$ – iron feliks May 16 '18 at 5:03
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    $\begingroup$ @SimonHenry This is not true for all Reedy categories, only for those with "cofibrant constants" and $\Delta$ happens not to be one. (See ncatlab.org/nlab/show/Reedy+category+with+fibrant+constants) $\endgroup$ – Karol Szumiło May 16 '18 at 6:15
  • $\begingroup$ @KarolSzumiło. You're right, I forgot about that ! Thanks for pointing it out. $\endgroup$ – Simon Henry May 16 '18 at 7:19

I'll just address the second question. It's not enough for the maps $X_i \to X_j$ to be fibrations, it's not even enough for them to be identity maps on a Kan complex! Take $X$ to be a Kan complex, $G$ to be a group and consider the trivial $G$ action on $X$ as a diagram whose shape is the one-object category $G$. Then the fixed points, $\lim_G X$, are just $X$, but the homotopy fixed points, $\mathrm{holim}_G X$, have the homotopy type of $\mathrm{Map}(BG, |X|)$.

So any reasonable answer to the first question must take into account the diagram shape, not just the individual simplicial sets $X_i$ and the maps $X_i \to X_j$.

  • $\begingroup$ @ Omar Antolin-Camarena. Yes, one must take into the account diagram shape (in my case the shape is given by cosimplicial category). The question in original form had been precisely about such shape but it got edited out. $\endgroup$ – iron feliks May 16 '18 at 4:53

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