# Homotopy limits of Simplicial sets

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 $\Delta$.

• 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. – Simon Henry May 16 '18 at 5:00
• @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. – iron feliks May 16 '18 at 5:03
• @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) – Karol Szumiło May 16 '18 at 6:15
• @KarolSzumiło. You're right, I forgot about that ! Thanks for pointing it out. – 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 $X_i \to X_j$.