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
A model category structure on the category Operad of Set-enriched coloured symmetric operads which generalizes the canonical model structure on Cat.
Call a morphism of operads $f : P \to Q$ a weak equivalence if
its underlying functor of categories is an essentially surjective functor;
for every collection $(c_1, \cdots, c_n; c)$ of colours it induces an isomorphism
(the operadic analog of being full and faithful).
Call a morphism $f : P \to Q$ a fibration if for every isomorphism in $Q$ and a lift of its source object to $P$ there is an isomorphism in $P$ covering it under $f$.
Call a morphism a cofibration if it is an injection on objects (on colours)
This defines a cofibrantly generated model category structure on Operad.
This is due to (Weiss 07).