Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
A model category structure on cosimplicial objects in unital, commutative algebras over some field .
Under the monoidal Dold-Kan correspondence this is Quillen equivalent to the model structure on commutative non-negative cochain dg-algebras.
The model structure
Let be a field of characteristic zero.
Say that morphism in (def. 1) is
- a weak equivalence if its image under the comparison functors from remark 1 is a quasi-isomorphism in ;
1.a fibration if is an epimorphism (i.e. degreewise a surjection).
this defines a model category structure, to be called the projective model structure on comsimplicial commutative -algebras. .
this is a cofibrantly generated model category
and a simplicial model category.
e.g. Toën 00, theorem 2.1.2
Simplicial model category structure
There is also the structure of an sSet-enriched category on (def. 2)
For a simplicial set and let be the corresponding -valued cochains on simplicial sets
See also Castiglioni-Cortinas 03, 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:
Toën 00, theorem 2.1.2
Relation to the model structure on cochain dgc-algebras
Under the monoidal Dold-Kan correspondence this is related to the model structure on commutative non-negative cochain dg-algebras.
Details are in
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:math/0306289)
There also aspects of relation to the model structure on dg-algebras is discussed. (See monoidal Dold-Kan correspondence for more on this).