nLab cosimplicial algebra



Higher algebra

Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Schanuel's lemma

Homology theories




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)C^\bullet(A) of a cosimplicial algebra AA 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)C^\bullet(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

and def 9.1, p. 18 of

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


  • As cosimplcial algebras are dual to simplicial spaces, each simplicial space XX 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:

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.

Last revised on July 14, 2021 at 11:03:36. See the history of this page for a list of all contributions to it.