Abelian objects in categories with normal projections

Abstract

It is known that in (regular) unital and in subtractive categories, internal abelian groups are simply behaved; e.g., they are the same as internal algebras $(A,s)$ satisfying $s(x,0)=x$ and $s(x,x)=0$, i.e., \emph{subtraction algebras}. Moreover, in these categorical settings, such internal abelian group structures are unique, and every morphism between the underlying objects of internal abelian groups is necessarily a morphism of internal abelian groups. It is also known that both (regular) unital and subtractive categories have normal projections, i.e., the isomorphism formula $(X\times Y)/Y\approx X$ holds. In this paper, we show that all properties of simple behaviour of internal abelian groups in unital and subtractive categories lift to arbitrary categories having normal projections