nLab cosimplicial algebra

Contents

Context

Higher algebra

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Schanuel's lemma

Homology theories

Theorems

Contents

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)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)

Examples

  • 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.

References

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.