A comonadic interpretation of Baues-Ellis homology of crossed modules

TL;DR

This paper develops a homology theory for crossed modules using comonadic methods, generalizing previous definitions and providing new formulas for their homological properties.

Contribution

It introduces a comonadic interpretation of Baues-Ellis homology, connecting it to Barr-Beck homology and deriving Hopf formulae in all dimensions.

Findings

Homology groups of crossed modules are characterized as Barr-Beck comonadic homology.

The approach recovers and generalizes Baues and Ellis's homology.

Hopf formulae are established for all homological dimensions.

Abstract

We introduce and study a homology theory of crossed modules with coefficients in an abelian crossed module. We discuss the basic properties of these new homology groups and give some applications. We then restrict our attention to the case of integral coefficients. In this case we regain the homology of crossed modules originally defined by Baues and further developed by Ellis. We show that it is an instance of Barr-Beck comonadic homology, so that we may use a result of Everaert and Gran to obtain Hopf formulae in all dimensions.