Homotopy limits of homotopically constant diagrams over contractible categories

Question

I suspect that the following result should be true and more or less well known:

Let $\mathcal{M}$ be a model category and $I$ a small category with contractible nerve. For every diagram $X: I \to \mathcal{M}$ and every $i \in I$, the natural map $holim_I X \to X(i)$ is a weak equivalence.

Indeed this is a direct generalization of 9.10 in [D.M. Kan et al. A classification theorem for diagrams of simplicial sets], where the result is stated for $\mathcal{M}$ the model category of simplicial sets. Has the general result been written up? I couldn't find anything in Hirschhorn or Hovey.

Answer 1

It is true. You can reduce to the case of simplicial sets by using the fact that $\mathbf{R} \mathrm{Hom} (T, -)$ preserves homotopy limits and (allowing $T$ to vary) is jointly conservative. Finding where this is written down is another matter altogether...

I'll just sketch something more concrete for the case of a simplicial model category $\mathcal{M}$. Let $X : \mathcal{I} \to \mathcal{M}$ be any diagram such that each $X i$ is fibrant. Using the cobar construction of homotopy limits (see e.g. Chapter 18 in [Hirschhorn]), it is immediate that $$\underline{\mathcal{M}} (T, \operatorname{holim} X) \cong \operatorname{holim} \underline{\mathcal{M}} (T, X)$$ for all cofibrant $T$ in $\mathcal{M}$. On the other hand, $\underline{\mathcal{M}} (T, Z)$ is a model of $\mathbf{R} \mathrm{Hom} (T, Z)$ when $T$ is cofibrant and $Z$ is fibrant. Thus, $\operatorname{holim} X \to X i$ is a weak equivalence in $\mathcal{M}$ if and only if $$\operatorname{holim} \underline{\mathcal{M}} (T, X) \to \underline{\mathcal{M}} (T, X i)$$ is a weak homotopy equivalence for every cofibrant object $T$ in $\mathcal{M}$. Hence, the claim reduces to the case where $\mathcal{M} = \mathbf{sSet}$.

The same strategy can be used for many other questions about homotopy (co)limits.