model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
(also nonabelian homological algebra)
For $\Delta$ the simplex category the functor category $sSet^{\Delta}$ is that of cosimplicial objects in simplicial sets: cosimplicial simplicial sets.
There are various standard model category structures on this category. The Reedy model structure is discussed in (BousfieldKan), the injective structure is discussed in (Jardine).
The totalization of a cosimplicial simplicial set $X^\bullet$ coincides with the sSet-enriched hom-object
where $\Delta : [k] \mapsto \Delta[k]$ is the canonical cosimplicial simplicial set given by the simplex-assignment.
Since $\Delta$ is cofibrant in the Reedy model structure it follows that totalization of Reedy-fibrant cosimplicial simplicial sets preserves weak equivalences. The following lists situations in which totalization respects weak equivalences even without this assumption.
Totalization is closely related to descent objects. If $A$ is a simplicial presheaf and $Y \to X$ is a hypercover, then the descent object is the sSet-enriched hom
If we decompose
into its cells by a coend, where now each $Y_n$ is a Set-valued presheaf (see co-Yoneda lemma), then this is
where the equality signs are isomorphisms of simplicial sets, the outside integral sign denotes the end, and in the integrand we are using that simplicial presheaves are simplicially enriched and tensored over simplicial sets.
So a standard class of examples of cosimplicial simplicial sets to keep in mind are those obtained by evaluating a simplicial presheaf degreewise on the components of a hypercover. Its totalization then is the corresponding descent object.
For $G^\bullet \to H^\bullet$ a morphism of cosimplicial groupoids which is degreewise an equivalence, also the induced morphism of totalizations
is a weak equivalence (of simplicial sets).
This is (Jardine, corollary 12).
Let $\Delta_+ \hookrightarrow \Delta$ be the subcategory of the simplex category on the co-face maps. Write $rTot$ for the corresponding totalization, called the restricted totalization.
For $G^\bullet \to H^\bullet$ a degreewise weak equivalence of strict 2-groupoids, the resulting morphism of connected components of restricted totalizations
is a weak equivalence.
This is (Prezma, theorem 6.1).
Retsricted to $\pi_0$ this statement appeared as (Yekutieli, theorem 2.4). Notice that it is indeed necessary to use the restricted totalization instead of the ordinary totalization here.
The Reedy model structure on $sSet^{\Delta}$ is discussed in Chapter X of
The injective model structure is discussed in
Totalization of cosimplicial strict 2-groupoids is considered in
and