Abelian objects in categories with normal projections

TL;DR

This paper extends the understanding of internal abelian groups from well-behaved categories like unital and subtractive categories to more general categories with normal projections, showing that their simple properties persist.

Contribution

It demonstrates that properties of internal abelian groups in unital and subtractive categories also hold in broader categories with normal projections.

Findings

Internal abelian groups behave simply in categories with normal projections

Uniqueness of internal abelian group structures is preserved

Morphisms between internal abelian groups are necessarily group homomorphisms

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