D. Bourn, G. Janelidze
As observed by J. Beck, and as we know from M. Barr's and his joint work on triple cohomology, the classical isomorphism Opext $\cong H^2$ that describes group extensions with abelian kernels, can be deduced from the equivalence between such extensions and torsors (in an appropriate sense). The same is known for many other "group-like"' algebraic structures, and now we present a purely-categorical version of that equivalence, essentially by showing that all torsors are extensions with abelian kernels in any pointed protomodular category, and by giving a necessary and sufficient condition for the converse.
Protomodular category, semi-abelian category, Opext functor, extension, torsor.
MSC 2000: 18G50, 18D35, 20J06, 18C10, 18G60