nLab model structure on dg-comodules

Idea

A model category structure on the category of comodules in chain complexes over a dg-coalgebra.

Details

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).

Related concepts

• model structure on dg-coalgebras

• model structure on coalgebras over a comonad

References

• 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)