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cosimplicial algebra

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Higher algebra

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

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 section 2.1 of

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.

Revised on February 24, 2011 08:13:44 by Urs Schreiber (89.204.137.99)