Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
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 model category structure
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 epimorphism (degreewise surjection).
This defines the projective model category structure on .
The simplicial model category structure
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
See also CasCor, p. 10.
For define the sSet-hom-object by
Let be a morphism of cosimplicial algebras and write
for the component of in degree with values in the copy of functions on the unique non-degenerate -simplex of . The coface maps obtained as the pullback of the face inclusions restrict on the non-degenerate -cells to the projections .
Accordingly, from the naturality squares for
the bottom horizontal morphism is fixed to have components
in the functions on the non-degenerate simplices.
By analogous reasoning this fixes all the components of in all lower degrees with values in the functions on degenerate simplices.
The above sSet-enrichment makes into a simplicially enriched category which is tensored and cotensored over .
And this is compatible with the model category structure:
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
The generalization to arbitrary cosimplicial rings is proposition 9.2 of
- J.L. Castiglioni G. Cortiñas, Cosimplicial versus DG-rings: a version of the Dold-Kan correspondence (arXiv)
There also aspects of relation to the model structure on dg-algebras is discussed. (See monoidal Dold-Kan correspondence for more on this).