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
2-natural transformation?
Every strict 2-category $K$ with finite strict 2-limits and finite strict 2-colimits becomes a model category (or, rather, its underlying 1-category does) in a canonical way, where:
The weak equivalences are the equivalences.
The fibrations are the morphisms that are representably isofibrations, i.e. the morphisms $e\to b$ such that $K(x,e)\to K(x,b)$ is an isofibration for all $x\in K$.
The cofibrations are determined.
We call it the 2-trivial model structure, as it is a 2-categorical analogue of the trivial model structure on any 1-category. It can be said to regard $C$ as an (∞,1)-category with only trivial k-morphisms for $k \geq 3$.
Every object is fibrant and cofibrant.
In Cat, this produces the canonical model structure.
By duality, any such category has another model structure, with the same weak equivalences but where the cofibrations are the iso-cofibrations and the fibrations are determined. In $Cat$, the two model structures are the same.
If $T$ is an accessible strict 2-monad on a locally finitely presentable strict 2-category $K$. Then the category $T Alg_s$ of strict $T$-algebras admits a transferred model structure from the 2-trivial model structure on $K$. The cofibrant objects therein are the flexible algebra?s.