Homotopy Limits over Fibered Categories

Question

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.

Answer 1

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.

Answer 2

The axiomatic way to present this is Heller's theory of homotopy theories, which is the same as Grothendieck's theory of derivators (see the references in the links below). As any model category defines a derivator (see here), what you want is precisely Lemma 2.3 in this paper.