Suppose I have a small category $ \mathcal{C} $ which is fibered over some category $\mathcal{I}$ in the categorical sense. That is, there is a functor $\pi : \mathcal{C} \rightarrow \mathcal{I}$ which is a fibration of categories. (One way to say this, I guess, is that $\mathcal{C}$ has a factorization system consisting of vertical arrows, i.e. the ones that $\pi$ sends to an identity arrow in $\mathcal{I}$, and horizontal arrows, which are the ones it does not. But there are many other characterizations.)
Now let $F : \mathcal{C} \rightarrow s\mathcal{S}$ be a diagram of simplicial sets indexed by $\mathcal{C}$. My question concerns the homotopy limit of $F$. Intuition tells me that there should be an equivalence
$$ \varprojlim_{\mathcal{C}} F \simeq \varprojlim_{\mathcal{I}} \left (\varprojlim_{\mathcal{C}_i} F_i \right ) $$
where I write $\mathcal{C}_i = \pi^{-1}(i)$ for any $i \in \mathcal{I}$, $F_i$ for the restriction of $F$ to $\mathcal{C}_i$ and $\varprojlim$ for the homotopy limit.
Intuitively this says that when $\mathcal{C}$ is fibered over $\mathcal{I}$, I can find the homotopy limit of a $\mathcal{C}$ diagram of spaces by first forming the homotopy limit of all the fibers, realizing that this collection has a natural $\mathcal{I}$ indexing, and then taking the homotopy limit of the resulting diagram.
Does anyone know of a result like this in the model category literature?
Update: After reading the responses, I was able to find a nice set of exercises here which go through this result in its homotopy colimit version.
Best Answer
I can't think of a reference for this. But here is what I would do:
Given any functor $\pi\colon C\to I$ (not necessarily fibered), there's a "homotopy right Kan extension" functor $$\lim{}^\pi \colon Func(C,sS) \to Func(I,sS),$$ and a weak equivalence $\lim_C = \lim_I \lim{}^\pi$. There's a formula to compute $\lim{}^\pi$ in terms of ordinary homotopy limits on comma categories; it looks like $$(\lim{}^\pi F)(i) = \lim{}_{i/\pi} F_i',$$ where $i/\pi$ is the comma category with objects $(c, f:i\to \pi c)$ where $c$ is an object of $C$. The functor $F_i'$ is the composite of $F$ with the forgetful functor $(i/\pi)\to C$.
I suspect that in your fibered category case, you are able to show that for each $i$ the evident inclusion $C_i\to (i/\pi)$ is "final" (or, possibly, "cofinal"; I can never remember which is which), and thus that $\lim_{i/\pi} F' = \lim_{C_i} F_i$.
The "yellow monster" of Bousfield-Kan is a reference for (co)finality condition in context of homotopy (co)limits. It may also discuss homotopy Kan extensions, though I'd have to check.