SEMI-ABELIAN CATEGORIES

In a seminal paper published in 1950, S. Mac Lane set the aim of developing a categorical treatment of groups, and began doing so in the more symmetrical abelian case. This has led to the theory of abelian categories, almost fully developed by the mid sixties. These categories cover, among others, abelian groups but not groups in general. It has become clear that the restriction to the abelian case is not a mere technical simplification. There have been attempts for a categorical treatment of certain aspects of groups: isomorphism theorems, commutator theory, non-abelian homological algebra, radical theory, semidirect products, but without a common background for them. Semi-abelian categories, introduced in a recent work of Janelidze, Marki, and Tholen, yield a framework for all these (and more) investigations, achieving thereby the task set by Mac Lane in 1950. The present talk will outline two approaches to semi-abelian categories.