Protoadditive functors, derived torsion theories and homology
arXiv:1111.5448 · doi:10.1016/j.jpaa.2014.12.015
Abstract
Protoadditive functors are designed to replace additive functors in a non-abelian setting. Their properties are studied, in particular in relationship with torsion theories, Galois theory, homology and factorisation systems. It is shown how a protoadditive torsion-free reflector induces a chain of derived torsion theories in the categories of higher extensions, similar to the Galois structures of higher central extensions previously considered in semi-abelian homological algebra. Such higher central extensions are also studied, with respect to Birkhoff subcategories whose reflector is protoadditive or, more generally, factors through a protoadditive reflector. In this way we obtain simple descriptions of the non-abelian derived functors of the reflectors via higher Hopf formulae. Various examples are considered in the categories of groups, compact groups, internal groupoids in a semi-abelian category, and other ones.
References in corpus (4)
Cited by in corpus (22)
- Relative Commutator Theory in Semi-Abelian Categories
- A torsion theory in the category of cocommutative Hopf algebras
- Resolutions, higher extensions and the relative Mal'tsev axiom
- Pretorsion theories in general categories
- Monotone-light factorisation systems and torsion theories
- Higher central extensions and cohomology
- A description of the fundamental group in terms of commutators and closure operators
- The Yoneda isomorphism commutes with homology
- Torsion theories and coverings of preordered groups
- Semi-localizations of semi-abelian categories
- Higher commutator conditions for extensions in Mal'tsev categories
- Universal central extensions of internal crossed modules via the non-abelian tensor product
- Conditional flatness, fiberwise localizations, and admissible reflections
- A comonadic interpretation of Baues-Ellis homology of crossed modules
- Hopf formulae for homology of skew braces
- Some remarks on protolocalizations and protoadditive reflections
- A classification theorem for normal extensions
- Non-additive derived functors via chain resolutions
- Some aspects of semi-abelian homology and protoadditive functors
- Pretorsion theories in prenormal categories
- Action representability of the category of internal groupoids
- Double groupoids and -groupoids in regular Mal'tsev categories