on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
Extra axioms on a monoidal model category $\mathbf{S}$ that guarantee a particularly good homotopy theory of $\mathbf{S}$-enriched categories are referred to as excellent model category structure (Lurie).
Let $\mathbf{S}$ be a monoidal model category. It is called excellent if
it is a combinatorial model category;
every monomorphism if a cofibration
the collection of cofibrations is closed under products;
it satisfies the invertibility hypothesis: inverting a morphism in an $\mathbf{S}$-enriched category does not change the homotopy type if the morphism was already a homotopy equivalence (…).
This is (Lurie, def. A.3.2.16).
Section A.3 in