Let $\mathcal{C}$ be a model category with a forgetful functor towards simplicial sets, and such that fibrations and trivial cofibrations are those whose underlying map on simplicial sets is a fibration, respectively trivial fibration.
Is there a criterion to decide whether formation of simplicial homotopy groups commutes with filtered colimits?



Commutation of homotopy groups with filtered colimits
I'm not aware of any such criterion (and I'm doubtful a very general one exists, if any), so this is not meant to be an answer (but for some reason I'm unable to post it as a comment right now).


A few remarks.
The question is slightly ill posed. What do you mean by "**commutation** of simplicial homotopy groups with filtered colimits"?
You probably mean formation of simplicial homotopy groups turns, in degree zero, filtered colimits in $\mathcal{C}$ into filtered colimits of pointed sets, and in positive degree it turns filtered colimits in $\mathcal{C}$ into filtered colimits of groups.


It seems to me a **necessary** condition on $\mathcal{C}$ is that it is **proper** (see Goerss-Jardine, Ch. II, $\S$8).
Indeed, if formation of higher simplicial homotopy groups commutes with filtered colimits in the above sense, then in particular pushouts along cofibrations preserve weak equivalences.


In fact, a sharper necessary condition is that weak equivalences in $\mathcal{C}$ are closed under filtered colimits.
This should be seen as a strong "finiteness condition" on $\mathcal{C}$: it is typically met if generating cofibrations and generating trivial cofibrations are small relative to the whole category $\mathcal{C}$ (which is stronger than requiring $\mathcal{C}$ to be "finitely generated").


You may then ask whether it is the case that in a model category satisfying your stated assumptions, and in which, in addition, filtered colimits preserve weak equivalences, formation of higher simplicial homotopy groups commutes with filtered colimits in the above sense.


I don't have a counterexample off the top of my head, but I'm led to believe this is not the case in general.


