A functorial treatment of factorization structures is presented, under extensive use of well-pointed endofunctors. Actually, so-called weak factorization systems are interpreted as pointed lax indexed endofunctors, and this sheds new light on the correspondence between reflective subcategories and factorization systems. The second part of the paper presents two important factorization structures in the context of pointed endofunctors: concordant-dissonant and inseparable-separable.

We develop a new approach to Commutator theory based on the theory of internal categorical structures, especially of so called pseudogroupoids. It is motivated by our previous work on internal categories and groupoids in congruence modular varieties.

. Each full reflective subcategory X of a finitely-complete category C gives rise to a factorization system (E; M) on C, where E consists of the morphisms of C inverted by the reflexion I : C ! X . Under a simplifying assumption which is satisfied in many practical examples, a morphism f : A ! B lies in M precisely when it is the pullback along the unit jB : B ! IB of its reflexion If : IA ! IB; whereupon f is said to be a trivial covering of B. Finally, the morphism f : A ! B is said to be a covering of B if, for some effective descent morphism p : E ! B, the pullback p f of f along p is a trivial covering of E. This is the absolute notion of covering; there is also a more general relative one, where some class \Theta of morphisms of C is given, and the class Cov(B) of coverings of B is a subclass -- or rather a subcategory -- of the category C #B ae C=B whose objects are those f : A ! B with f 2 \Theta. Many questions in mathematics can be reduced to asking whether Cov(B) is re...

and Ellis’s higher Hopf formulae for homology of groups to arbitrary semiabelian monadic categories. Given such a category A and a chosen Birkhoff subcategory B of A, thus we describe the Barr-Beck derived functors of the reflector of A onto B in terms of centralization of higher extensions. In case A

Relationship is clarified between the notions of linear extension of algebraic theories, and central extension, in the sense of commutator calculus, of their models. Varieties of algebras turn out to be nilpotent Maltsev precisely when their theories may be obtained as results of iterated linear extensions by bifunctors from the so called abelian theories. The latter theories are described; they are slightly more general than theories of modules over a ring.

Extending the work of Fröhlich, Lue and Furtado-Coelho, we consider the theory of Baer invariants in the context of semi-abelian categories. Several exact sequences, relative to a subfunctor of the identity functor, are obtained. We consider a notion of commutator which, in the case of abelianization, corresponds to Smith's. The resulting notion of centrality fits into Janelidze and Kelly's theory of central extensions. Finally we propose a notion of nilpotency, relative to a Birkhoff subcategory of a semiabelian category.

Abstract. We show that every algebraically–central extension in a Mal’tsev variety — that is, every surjective homomorphism f: A− → B whose kernel–congruence is contained in the centre of A, as defined using the theory of commutators — is also a central extension in the sense of categorical Galois theory; this was previously known only for varieties of Ω-groups, while its converse is easily seen to hold for any congruence–modular variety. 1.

By defining a closure operator on effective equivalence relations in a regular category C, it is possible to establish a bijective correspondence between these closure operators and the regular epireflective subcategories L of C. When C is an exact Goursat category this correspondence restricts to a bijection between the Birkhoff closure operators on effective equivalence relations and the Birkhoff subcategories of C. In this case it is possible to provide an explicit description of the closure, and to characterise the congruence distributive Goursat categories.

This article treats the problem of deriving the reflector of a semi-abelian

For a given Galois structure on a category C and an effective descent morphism p : E!B in C we describe the category of so-called weakly split objects over (E; p) in terms of internal actions of the Galois (pre)groupoid of (E; p) with an additional structure. We explain that this generates various known results in categorical Galois theory and in particular two results of M. Barr and R. Diaconescu [BD]. We also give an elaborate list of examples and applications.