Abelian objects in categories with normal projections

TL;DR

This work shows that the robust behavior of internal abelian groups in regular unital and subtractive categories extends to any category with normal projections. It establishes a subtractive theory of abelian objects where a unique internal subtraction induces a group structure, and morphisms automatically preserve abelian operations; central to this are the equivalences surrounding the isomorphism law $(A\times B)/B \cong A$ and the fundamental identity $s(s(x,z),s(y,z))=s(x,y)$. The results apply beyond the classical settings to congruence hyperextensible and centralic categories, demonstrating that normal projections provide a broad crystallographic framework for abelian objects. Together, these findings significantly generalize internal abelian-group theory across a wide class of pointed categories, enabling uniform treatment of abelian structures via 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

Theorems & Definitions (7)

• Lemma 1.1
• Theorem 1.2
• proof
• Remark 1.3
• Theorem 2.1
• proof
• Theorem 2.2