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
An enriched model category is an enriched category $C$ together with the structure of a model category on the underlying category $C_0$ such that both structures are compatible in a reasonable way.
Let $V$ be a monoidal model category.
A $V$-enriched model category is
an V-enriched category $C$
with the structure of a model category on the underlying category $C_0$
such that
for every cofibration $i \colon A \to B$ in $V_0$ and every fibration $p \colon X \to Y$ in $C_0$ the pullback-powering morphism (dual to the pushout product) with respect to the powering in $V$
is a fibration with respect to the model structure on $V$;
and is an acyclic fibration whenever $i$ or $p$ are acyclic.
The last two conditions here are equivalent to the fact that the copower
is a Quillen bifunctor.
(…)
The most familiar examples of enriched model categories are simplicial model categories.
For a pointed model category or even stable model category there is enrichment in pointed simplicial sets with the smash product as tensor product.