Let $\mathcal{M}$ be a
locally finitely presentable model category, cofibrantly generated by two sets $\mathcal{I}$ and $\mathcal{J}$ of cofibrations and trivial cofibrations with presentable domain and codomain.
I know that weak equivalences and fibrations are stable by filtered colimits.
What can be said about cofibrations and trivial cofibrations?
Is there a class of good examples in which this is known to be true?
Are there additional axioms that can be imposed that ensure this?
If both cofibrations and weak equivalences are stable under filtered colimits, then so are trivial cofibrations. This happens for instance if $\mathcal{M}$ is a presheaf category on an elegant Reedy category (such as $\Delta$) with cofibrations the monomorphisms, whatever the weak equivalences are (see Cor. 3.4.41 & (the proof of) Prop. 8.2.9 here). This is also true for cochain complexes in a Grothendieck abelian category (with cofibrations as monomorphisms and quasi-isomorphisms as weak equivalences). Similarly, if you consider simplicial sheaves on a site. More generally, if you can define cofibrations and weak equivalences using functors which commute with filtered colimits (e.g. suitable kernels to detect monomorphisms in a topos, cohomology sheaves or sheaves of homotopy groups to detect weak equivalences), you have a good chance of survival. This kind of properties is easily seen to be preserved under left Bousfield localizations.
