symmetric monoidal (∞,1)-category of spectra
(also nonabelian homological algebra)
A cosimplicial algebra – similarly a cosimplicial ring – is a cosimplicial object in the category of algebras (of rings).
Under the monoidal Dold–Kan correspondence, cosimplicial algebras are essentially identified with differential graded algebras in non-negative degree: the Moore cochain complex $C^\bullet(A)$ of a cosimplicial algebra $A$ is a differential graded algebra where the degreewise product on the cosimplicial algebra maps to the cup product operation that gives the monoid structure $C^\bullet(A)$.
A standard model category structure on the category of cosimplicial rings is the following
fibrations are the degreewise surjections
weak equivalences are the morphisms that induce isomorphisms in cohomotopy
cofibrations are defined by their left lifting property.
For more see model structure on cosimplicial algebras.
References are section 2.1 of
and def 9.1, p. 18 of
As cosimplcial algebras are dual to simplicial spaces, each simplicial space $X$ gives rise to a cosimplicial algebra of functions on it. A list of examples is given at Chevalley-Eilenberg algebra.
The model category structure on cosimplicial algebras is discussed in detail in section 2.1 of
The Quillen equivalence between cosimplicial algebras and cochain dg-algebras is discussed in
A bit about cosimplicial algebras is in section 7 of
This also discusses aspects of their image in dg-algebras under the Moore complex-functor. See monoidal Dold-Kan correspondence for more on that.