On satellites in semi-abelian categories: Homology without projectives

Abstract

Working in a semi-abelian context, we use Janelidze's theory of generalised satellites to study universal properties of the Everaert long exact homology sequence. This results in a new definition of homology which does not depend on the existence of projective objects. We explore the relations with other notions of homology, and thus prove a version of the higher Hopf formulae. We also work out some examples.

29 pages; major changes in Example 4.15, minor changes throughout the text

References in corpus (4)

• Higher Hopf formulae for homology via Galois Theory
• On the second cohomology group in semi-abelian categories
• Higher central extensions and Hopf formulae
• On satellites in arbitrary categories

Cited by in corpus (10)

• Resolutions, higher extensions and the relative Mal'tsev axiom
• Higher central extensions and cohomology
• A description of the fundamental group in terms of commutators and closure operators
• Higher Central Extensions in Mal'tsev Categories
• The Yoneda isomorphism commutes with homology
• Galois theory and commutators
• A comonadic interpretation of Baues-Ellis homology of crossed modules
• Some aspects of semi-abelian homology and protoadditive functors
• A "working mathematician's" definition of semi-abelian categories
• The fundamental group functor as a Kan extension