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
The standard model category structure on cosimplicial objects in unital, commutative algebras over $k$.
Under the monoidal Dold-Kan correspondence this is Quillen equivalent to the model structure on commutative non-negative cochain dg-algebras.
Write $Alg_k^\Delta$ for the category of cosimplicial objects in the category of unital, commutative $k$-algebras. Sending algebras to their underlying $k$-modules yields a forgetful functor
The Dold-Kan correspondence provides the normalized cochain complex (Moore complex) functor
Define a morphism $f : A \to B$ of cosimplicial algebras is a morphism is a weak equivalence if
is a quasi-isomorphism in $Ch^\bullet_+(k)$.
Say a morphism of cosimplicial algebras is a fibration if it is a epimorphism (degreewise surjection).
This defines the projective model category structure on $Alg_k^\Delta$.
There is also the structure of an sSet-enriched category of $Alg_k^\Delta$.
For $X$ a simplicial set and $A \in Alg_k$ let $A^X \in Alg_k^\Delta$ be the corresponding $A$-valued cochains on simplicial sets
If we write $C(X) := Hom_{Set}(X_\bullet,k)$ for the cosimplicial algebra of cochains on simplicial sets then for $X$ degreewise finite this may be written as
where the tensor product is the degreewise tensor product of $k$-algebras.
See also CasCor, p. 10.
For $A,B \in Alg_k^\Delta$ define the sSet-hom-object $Alg_k^\Delta(A,B)$ by
For $B \in Alg_k$ regarded as a constant cosimplicial object under the canonical embedding $Alg_k \hookrightarrow Alg_k^\Delta$ we have
Let $f : A \to B \otimes C(\Delta[n])$ be a morphism of cosimplicial algebras and write
for the component of $f$ in degree $n$ with values in the copy $B = B \otimes k$ of functions $k$ on the unique non-degenerate $n$-simplex of $\Delta[n]$. The $n+1$ coface maps $C(\Delta[n])_n \leftarrow C(\Delta[n])_{n-1}$ obtained as the pullback of the $(n+1)$ face inclusions $\Delta[n-1] \to \Delta[n]$ restrict on the non-degenerate $(n-1)$-cells to the $n+1$ projections $k \leftarrow k^{n+1} : p_i$.
Accordingly, from the naturality squares for $f$
the bottom horizontal morphism is fixed to have components
$f_{n-1} = (f_n \circ \delta_0, \cdots, f_n \circ \delta_n)$
in the functions on the non-degenerate simplices.
By analogous reasoning this fixes all the components of $f$ in all lower degrees with values in the functions on degenerate simplices.
The above sSet-enrichment makes $Alg_k^\Delta$ into a simplicially enriched category which is tensored and cotensored over $sSet$.
And this is compatible with the model category structure:
With the definitions as above, $Alg_k^\Delta$ 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).