The normalized Moore complex functor is usually considered taking simplicial abelian groups to chain complexes. But there is an obvious dual version that takes cosimplicial abelian groups to N-graded cochain complexes.
Moreover, applied on a cosimplicial group that has the structure of a cosimplicial ring, the Moore cochain complex yields a cochain complex that has the structure of a dg-algebras (the details are at monoidal Dold-Kan correspondence).
This seems obvious and useful enough, but I find surprisingly little literature on this dual monoidal Dold-Kan correspondence. In fact the only relevant reference that I am aware of is Castiglioni-Cortinas, Cosimplicial versus dg-rings: a version of the Dold-Kan correspondence.
But it would seem that instead considering directly the Moore cochain complex functor on cosimplicial rings would be at least as interesting.
I can dream up some of its properties myself, but I keep feeling I must be missing the canonical literature on this, which must exist. Does anyone have more references on the Moore cochain complex functor on cosimplicial rings/algebras?