on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
Let $Ab^{\Delta}$ be the category of cosimplicial objects in the category Ab of abelian groups – the category of cosimplicial abelian groups .
This entry discusses structures of model categories on $Ab^\Delta$.
By the dual Dold-Kan correspondence there is an equivalence of categories $Ab^\Delta \stackrel{\overset{\Xi}{\leftarrow}}{\underset{N}{\to}} Ch^\bullet_+(Ab)$ 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 $Ab^\Delta$.
Since Ab has all limits and colimits, the category of cosimplicial objects (as described there) $Ab^\Delta$ inherits canonically the structure of an sSet-enriched category which is powered and copowered.
Write $Ab^\Delta_{proj}$ for the model structure that is induced by the dual Dold-Kan correspondence $Ab^\Delta \simeq Ch^\bullet_+(Ab)$ 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 $Ab^\Delta_{proj}$ whose fibrations are also the degreewise surjections in $Ab$ (using that the normalized cochain complex-functor preserves surjections.)
The canonical $sSet$-enrichement of $Ab^\Delta$ is compatible with the model category structure $Ab^\Delta_{proj}$ in that the combination gives $Ab^\Delta$ the structure of a simplicial model category.
We need check the pushout-product axiom of an enriched model category of the standard model structure on simplicial sets $sSet_{Quillen}$
So we need to show that for $i : C \to C'$ any cofibration in $sSet_{Quillen}$ and $j : X \to Y$ a fibration of cosimplcial abelian groups (degreewise surjection) the morphism
induced by the powering $(-)^{(-)} : Ab^\Delta \times sSet \to Ab^\Delta$ is a fibration, which is acyclic if $i$ or $j$ is.
That $k$ is a fibration is easily checked. To see acyclicity we first notice the following
Lemma. If $i : C \to C'$ is a weak equivalence then for every cosimplicial abelian group $A$ we have $A^i$ is a weak equivalence.
To see this observe that $A^C$ is the diagonal of an evident bisimplicial abelian group and that $A^i$ 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 $f : (x,y) \mapsto j^C(x) - Y^i(y)$. This induces a long exact sequence in cohomology
If $i$ is a weak equivalence, then by the above lemma we have that
Inspection of the connecting homomorphism then shows that $H^p(Y^C) \to H^{p+1}(X^C \times_{Y^C} Y^{C'})$ is the 0-map. In total this implies that we have an isomorphism
for all $p$, and hence that
is a weak equivalence. Since by the above lemma also $X^i : X^{C'} \to X^C$ is a weak equivalence, it follows by 2-out-of-3 that the morphism $k$ is indeed a weak equivalence if $i$ is.
An analogous argument shows that $k$ is a weak equivalence if $j$ 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 $Ab^\Delta_{proj}$ 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.