on dg-algebras/on dg-coalgebras and on on cosimplicial rings (related by monoidal Dold-Kan correspondence)
The standard model category structure on cosimplicial objects in unital, commutative algebras over .
Under the monoidal Dold-Kan correspondence this is Quillen equivalent to the model structure on commutative non-negative cochain dg-algebras.
Write for the category of cosimplicial objects in the category of unital, commutative -algebras. Sending algebras to their underlying -modules yields a forgetful functor
The Dold-Kan correspondence provides the normalized cochain complex (Moore complex) functor
Define a morphism of cosimplicial algebras is a morphism is a weak equivalence if
is a quasi-isomorphism in .
Say a morphism of cosimplicial algebras is a fibration if it is a monomorphism (degreewise surjection).
This defines the projective model category structure on .
There is also the structure of an sSet-enriched category of .
For a simplicial set and let be the corresponding -valued cochains on simplicial sets
If we write for the cosimplicial algebra of cochains on simplicial sets then this may be written as
where the tensor product is the degreewise tensor product of -algebras.
For define the sSet-hom-object by
This makes into a simplicially enriched category which is tensored and cotensored over .
And this is compatible with the model category structure:
With the definitions as above, is a simplicial model category.
Under the monoidal Dold-Kan correspondence this is related to the model structure on commutative non-negative cochain dg-algebras.
Details are in section 2.1 of
See also
The generalization to arbitrary cosimplicial rings is proposition 9.2 of
There also aspects of relation to the model structure on dg-algebras is discussed. (See monoidal Dold-Kan correspondence for more on this).