related by the Dold-Kan correspondence
where is the canonical cosimplicial simplicial set given by the simplex-assignment.
Since 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.
If we decompose
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.
is a weak equivalence (of simplicial sets).
This is (Jardine, corollary 12).
is a weak equivalence.
This is (Prezma, theorem 6.1).
Retsricted to 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 is discussed in Chapter X of
The injective model structure is discussed in
Totalization of cosimplicial strict 2-groupoids is considered in