An important aspect of group theory is the study of normal subgroups. A protomodular category, even a non-pointed one, is defined in such a way that it possesses an intrinsic notion of normal subobject. The concept is due to Dominique Bourn and as such sometimes referred to as Bourn-protomodularity.
(Taken from Bourn04)
Consider any finitely complete category and denote by the category whose objects are the split epimorphisms in with a given splitting and morphisms the commutative squares between these data. Denote by the functor associating its codomain with any split epimorphism. Since the category has pullbacks, the functor is a fibration which is called the fibration of points.
Any map induces, by pullbacks, a change of base functor denoted between the fibres above and .
Certain categories of algebraic varieties, such as the category of groups, the category of rings, the category of associative or Lie algebras over a given ring , the category of Heyting algebras, the varieties of -groups. (It is shown in Bourn-Janelidze that a variety of universal algebras is protomodular if and only if it has -ary terms , binary terms , and -ary term satisfying the identities and for each .)
Constructions which inherit the property of being protomodular, such as the slice categories and the fibres of the fibration of pointed objects for instance, or more generally the domain of any pullback preserving and conservative functor ; when its codomain is protomodular.
The dual of a topos.
A pointed protomodular category is strongly unital, and
A category is strongly protomodular, when it is protomodular and such that any change of base functor is a normal functor, i.e. a left exact conservative functor which reflects the normal monomorphisms.
Francis Borceux, Dominique Bourn, Mal’cev, protomodular, homological and semi-abelian categories, Mathematics and Its Applications 566, Kluwer 2004
Dominique Bourn, Protomodular aspect of the dual of a topos, Advances in Mathematics 187(1), pp. 240-255, 2004.
Dominique Bourn, Action groupoid in protomodular categories, TAC
Dominique Bourn, George Janelidze, Characterization of protomodular varieties of universal algebras, (TAC)