nLab
cosimplicial algebra

Idea

A cosimplicial algebra – similarly a cosimplicial ring – is a cosimplicial object in the category of algebras (of rings).

Relation to differential graded algebras

Under the monoidal Dold–Kan correspondence, cosimplicial algebras are essentially identified with differential graded algebras in non-negative degree: the Moore cochain complex $C^{•}(A)$ 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 $C^{•}(A)$.

Model category structure

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

• Bertrand Toen, Affine stacks (Champs affines) (arXiv:math/0012219) .

and def 9.1, p. 18 of

• Castiglioni, Cortinas, Cosimplicial versus dg-rings (arXiv)

Examples

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.

References

The model category structure on cosimplicial algebras is discussed in detail in section 2.1 of

• Bertrand Toen, Affine stacks (Champs affines) (arXiv:math/0012219) .

The Quillen equivalence between cosimplicial algebras and cochain dg-algebras is discussed in

• Castiglioni, Cortinas, Cosimplicial versus dg-rings (arXiv)

A bit about cosimplicial algebras is in section 7 of

• José Burgos Gil, The regulators of Beilinson and Borel (pdf)

This also discusses aspects of their image in dg-algebras under the Moore complex-functor. See monoidal Dold-Kan correspondence for more on that.