The notion of semi-abelian category as proposed in this paper is designed to capture typical algebraic properties valid for groups, rings and algebras, say, just as abelian categories allow for a generalized treatment of abelian-group and module theory. In modern terms, semi-abelian categories are exact in the sense of Barr and protomodular in the sense of Bourn and have finite coproducts and a zero object. We show how these conditions relate to "old" exactness axioms involving normal monomorphisms and epimorphisms, as used in the fifties and sixties, and we give extensive references to the literature in order to indicate why semi-abelian categories provide an appropriate notion to establish the isomorphism and decomposition theorems of group theory, to pursue general radical theory of rings, and how to arrive at basic statements as needed in homological algebra of groups and similar non-abelian structures. Mathematics Subject Classification: 18E10, 18A30, 18A32. Key words:...

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.

This article shows that the distributive laws of Beck in the bicategory of sets and matrices determine strict factorization systems on their composite monads (=categories). Conversely, it is shown that strict factorization systems on categories give rise to distributive laws. Moreover, these processes are shown to be mutually inverse in a precise sense. Further, an extension of the distributive law concept provides a correspondence with the classical orthogonal factorization systems.

Abstract. For a category C we investigate the problem of when the coproduct ⊕ and the product functor ∏ from C I to C are isomorphic for a fixed set I, or equivalently, when the two functors are Frobenius functors. We show that for an Ab category C this happens if and only if the set I is finite. Moreover, this happens even in a much more general case, if there is a morphism in C that is invertible with respect to the addition of morphisms. If C does not have this property then we give an example to see that the two functors can be isomorphic for infinite sets I. However we show that ⊕ and ∏ are always isomorphic on a suitable subcategory of C I which is isomorphic to C I but is not a full subcategory. For the module category case we provide a different proof to display an interesting connection to the notion of Frobenius corings.

Abstract. Let C be a coalgebra. We investigate the problem of when the rational part of every finitely generated C ∗-module M is a direct summand M. We show that such a coalgebra must have at most countable dimension, C must be artinian as right C ∗-module and injective as left C ∗-module. Also in this case C ∗ is a left Noetherian ring. Following the classic example of the divided power coalgebra where this property holds, we investigate a more general type of coalgebras, the chain coalgebras, which are coalgebras whose lattice of left (or equivalently, right, two-sided) coideals form a chain. We show that this is a left-right symmetric concept and that these coalgebras have the above stated splitting property. Moreover, we show that this type of coalgebras are the only infinite dimensional colocal coalgebras for which the rational part of every finitely generated left C ∗-module M splits off in M, so this property is also left-right symmetric and characterizes the chain coalgebras among the colocal coalgebras.