TITLE: Homotopy limits of Simplicial sets
QUESTION [5 upvotes]: 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 $\Delta$.

REPLY [4 votes]: 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 $X_i \to X_j$.