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
In a model category, an object is said to be fibrant if the unique map to the terminal object is a fibration.
Dually, is said to be cofibrant if the unique map from the initial object is a cofibration.