model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
A model category structure on the category of comodules in chain complexes over a dg-coalgebra.
Let $C$ be a differential graded-cocommutative coalgebra over a field.
Model structure of the second kind
There exists a model category structure on the category $C dgCoMod$ of dg-comodules over $C$ whose
weak equivalences are the morphisms whose mapping cone is a co-acyclic comodule,
fibrations are the surjections whose kernel is such that its underlying comodule over the underlying coalgebra of $C$ is an injective object.
This is due to (Positelski 11, 8.2 Theorem (a)).
Model structure of the first kind
There is another model structure where the fibrations in addition satisfy the condition that their kernel $K$ satisfies that for all acyclic $N$, then $\underline{Hom}_C(N,K)$ is acyclic.
This is due to (Positelski 11, 8.2 Remark), there called the “model category structure of the first kind”. This is also reviewed as (Pridham 13, prop. 2.2).
Leonid Positselski, Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence, Memoirs of the Amer. Math. Soc. 212 (2011), no.996, (arXiv:0905.2621)
Jonathan Pridham, Tannaka duality for enhanced triangulated categories (arXiv:1309.0637)