Filtered colimit of fibrations

In a model category $\mathcal{C}$, is the filtered colimit of fibrations, resp. trivial fibrations, a fibration, resp. trivial fibration?

Thm. 1.2.3.5 in Toen-Vezzosi’s “Homotopical algebraic geometry, II” (https://arxiv.org/pdf/math/0404373.pdf) seems to give a criterion, but it points to the wrong reference, as nothing of the sort is in Hovey’s book “Model Categories”.

Answer

In Lemma 7.4.1 of Hovey’s book, he does prove that colimits of $\lambda$-sequences of cofibrations preserve fibrations, respectively trivial fibrations. However, upon inspecting the proof, the assumption on the transition maps being cofibrations is used only insofar domains and codomains of the generating cofibrations and trivial cofibrations are assumed to be finite relative to cofibrations. In Toen-Vezzosi, such domains and codomains are finite relative to the whole category, hence you can run the same proof as in Hovey, removing this assumption.

I don’t know of a general criterion about preservation of fibrations/trivial fibrations under a class of colimits, but I don’t expect such preservation to hold in great generality. If filtered colimits preserve fibrations and trivial cofibrations, then, upon endowing the diagram category with the induced model structure constructed in $\S$ 5 of Hovey’s book, $\text{colim}$ being a left Quillen functor (and hence already preserving cofibrations and trivial cofibrations) would come to preserve weak equivalences.

Simplicial commutative rings with the projective model structure is a counterexample: coproducts clearly don’t preserve weak equivalences (pick two ring maps with same source and that are not $\text{Tor}$-independent).

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