On the notion of a semi-abelian category in the sense of Palamodov

TL;DR

This paper explores the concept of semi-abelian categories in the sense of Palamodov, providing equivalent conditions, characterizations, and examples from functional analysis to illustrate their natural occurrence.

Contribution

It establishes multiple equivalent conditions for semi-abelianity, characterizes left and right semi-abelian categories, and presents functional analysis examples demonstrating their distinction.

Findings

Characterizations of semi-abelian categories via six equivalent properties

Distinction between left and right semi-abelian categories

Examples from functional analysis illustrating these notions

Abstract

In the sense of Palamodov, a preabelian category is semi-abelian if for every morphism the natural morphism between the cokernel of its kernel and the kernel of its cokernel is simultaneously a monomorphism and an epimorphism. In this article we present several conditions which are all equivalent to semi-abelianity. First we consider left and right semi-abelian categories in the sense of Rump and establish characterizations of these notions via six equivalent properties. Then we use these properties to deduce the characterization of semi-abelianity. Finally, we investigate two examples arising in functional analysis which illustrate that the notions of right and left semi-abelian categories are distinct and in particular that such categories occur in nature.