model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
A model category structure on cosimplicial objects in unital, commutative algebras over some field $k$.
Under the monoidal Dold-Kan correspondence this is Quillen equivalent to the model structure on commutative non-negative cochain dg-algebras.
Let $k$ be a field of characteristic zero.
Write $cAlg_k^\Delta$ for the category of cosimplicial objects in the category of unital, commutative algebras over $k$.
Sending $k$-algebras to their underlying $k$-modules yields a forgetful functor
from cosimplicial $k$-allgebras (def. ) to cosimplicial objects in $k$-vector spaces.
Moreover, the Dold-Kan correspondence provides the normalized cochain complex functor
from cosimplicial $k$-vector spaces to cochain complexes (i.e. with differential of degree +1) in non-negative degrees.
Say that morphism $f \colon A \to B$ in $cAlg_k^{\Delta}$ (def. ) is
1.a fibration if $f$ is an epimorphism (i.e. degreewise a surjection).
Then
this defines a model category structure, to be called the projective model structure on comsimplicial commutative $k$-algebras. $(cAlg_k^\Delta)_{poj}$.
this is a cofibrantly generated model category
and a simplicial model category.
The first two statements follow by observing that $(cAlg_k^{\Delta})_{proj} is$the transferred model structure along the forgetful functor $U \circ N$ from remark of the projective model structure on chain complexes, by this prop..
There is also the structure of an sSet-enriched category on $cAlg_k^\Delta$ (def. )
For $X$ a simplicial set and $A \in Alg_k^{\Delta}$ let $A^X \in Alg_k^\Delta$ be the corresponding $A$-valued cochains on simplicial sets
If we write $C(X) \coloneqq Hom_{Set}(X_\bullet,k)$ 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 Castiglioni-Cortinas 03, 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 $cAlg_k^\Delta$ 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, $(cAlg_k^\Delta)_{proj}$ 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
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).