36 research outputs found
Some aspects of semi-abelian homology and protoadditive functors
In this note some recent developments in the study of homology in
semi-abelian categories are briefly presented. In particular the role of
protoadditive functors in the study of Hopf formulae for homology is explained.Comment: 7 page
Protoadditive functors, derived torsion theories and homology
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
Monotone-light factorisation systems and torsion theories
Given a torsion theory (Y,X) in an abelian category C, the reflector I from C
to the torsion-free subcategory X induces a reflective factorisation system (E,
M) on C. It was shown by A. Carboni, G.M. Kelly, G. Janelidze and R. Par\'e
that (E, M) induces a monotone-light factorisation system (E',M*) by
simultaneously stabilising E and localising M, whenever the torsion theory is
hereditary and any object in C is a quotient of an object in X. We extend this
result to arbitrary normal categories, and improve it also in the abelian case,
where the heredity assumption on the torsion theory turns out to be redundant.
Several new examples of torsion theories where this result applies are then
considered in the categories of abelian groups, groups, topological groups,
commutative rings, and crossed modules.Comment: 12 page
The fundamental group functor as a Kan extension
We prove that the fundamental group functor from categorical Galois theory
may be computed as a Kan extension.Comment: Final published version. 26 pages. Dedicated to Rene Guitart on the
occasion of his sixty-fifth birthda
Galois theory and commutators
We prove that the relative commutator with respect to a subvariety of a
variety of Omega-groups introduced by the first author can be described in
terms of categorical Galois theory. This extends the known correspondence
between the Froehlich-Lue and the Janelidze-Kelly notions of central extension.
As an example outside the context of Omega-groups we study the reflection of
the category of loops to the category of groups where we obtain an
interpretation of the associator as a relative commutator.Comment: 14 page
Effective descent morphisms of regular epimorphisms
Let be a regular category with pushouts of regular epimorphisms by
regular epimorphism and the category of regular epimorphisms in .
We prove that every regular epimorphism in is an effective descent
morphism if, and only if, is a regular category. Then, moreover, every
regular epimorphism in is an effective descent morphism. This is the case,
for instance, when is either exact Goursat, or ideal determined, or is a
category of topological Mal'tsev algebras, or is the category of -fold
regular epimorphisms in any of the three previous cases, for any
Higher central extensions and Hopf formulae
Higher extensions and higher central extensions, which are of importance to
non-abelian homological algebra, are studied, and some fundamental properties
are proven. As an application, a direct proof of the invariance of the higher
Hopf formulae is obtained