October 1998
George Janelidze, Georgian Academy of Sciences (Tbilisi)
Semi-abelian categories

The well known notion of abelian category goes back essentially to S.Mac Lane's famous paper "Duality for groups"; these categories provide an abstract setting for the theory of (abelian groups and) modules. The categories which the authors decided to call semiabelian similarly generalize the varieties of groups (possibly with multiple operators, which includes rings and algebras). The axioms for semiabelianess are not new: to the contrary, we obtain a surprising theorem which in fact says that the various old and new attempts to define such a notion do agree. Our results once again show the power of two beautiful discovers: M.Barr's exactness and D.Bourn's protomodularity.