related by the Dold-Kan correspondence
This entry discusses structures of model categories on .
By the dual Dold-Kan correspondence there is an equivalence of categories with the category of cochain complexes in non-negative degree. Since Ab is an abelian category, we have by general results various model structures on cochain complexes. Via the Dold-Kan equivalence, all of these induce model structures on .
Write for the model structure that is induced by the dual Dold-Kan correspondence from the model structure on cochain complexes whose fibrations are the degreewise surjections (and weak equivalences the usual quasi-isomorphisms). This is described in detail here. So this induces the model structure whose fibrations are also the degreewise surjections in (using that the normalized cochain complex-functor preserves surjections.)
The canonical -enrichement of is compatible with the model category structure in that the combination gives the structure of a simplicial model category.
So we need to show that for any cofibration in and a fibration of cosimplcial abelian groups (degreewise surjection) the morphism
induced by the powering is a fibration, which is acyclic if or is.
That is a fibration is easily checked. To see acyclicity we first notice the following
Lemma. If is a weak equivalence then for every cosimplicial abelian group we have is a weak equivalence.
To see this observe that is the diagonal of an evident bisimplicial abelian group and that is then in one argument a degreewise quasi-isomorphism. Since forming total complexes preserves degreewise equivalences, the lemma follows.
To continue the main proof, notice that we have a short exact sequence
with . This induces a long exact sequence in cohomology
If is a weak equivalence, then by the above lemma we have that
for all , and hence that
is a weak equivalence. Since by the above lemma also is a weak equivalence, it follows by 2-out-of-3 that the morphism is indeed a weak equivalence if is.
An analogous argument shows that is a weak equivalence if is.
This argument is essentially that on page 41 of (Toën)
The model structure on cosimplicial algebras is discussed in detail in
The above proof that is a simplicial model category mimics the proof on page 41 there. Indeed, the claim is that the model structure on cosimplicial algebras is the transferred model structure induced by the above from the evident forgetful functor.