Salamander lemma for non-abelian group-like structures

TL;DR

This paper generalizes the salamander lemma from homological algebra to a broad class of non-abelian group-like structures within a self-dual axiomatic framework, extending its applicability.

Contribution

It establishes a generalized salamander lemma applicable to various non-abelian structures and axiomatic contexts, unifying and extending previous results.

Findings

Generalization of the salamander lemma to non-abelian structures

Application to semi-abelian categories and exact categories

Self-dual axiomatic framework for the lemma

Abstract

It is well known that the classical diagram lemmas of homological algebra for abelian groups can be generalized to non-abelian group-like structures, such as groups, rings, algebras, loops, etc. In this paper we establish such a generalization of the "salamander lemma" due to G. M. Bergman, in a self-dual axiomatic context (developed originally by Z. Janelidze), which applies to all usual non-abelian group-like structures and also covers axiomatic contexts such as semi-abelian categories in the sense of G. Janelidze, L. Marki and W. Tholen and exact categories in the sense of M. Grandis.