The Two-Square Lemma and the connecting morphism

TL;DR

This paper generalizes the Two-Square Lemma from abelian and preabelian categories and clarifies the nature of the connecting morphism in the Snake Lemma, contributing to category theory foundations.

Contribution

It extends the Two-Square Lemma to more general categories and establishes the equivalence of two definitions of the connecting morphism.

Findings

Generalization of the Two-Square Lemma to broader categories

Proof of equivalence of two definitions of the connecting morphism

Enhanced understanding of the Snake Lemma in category theory

Abstract

We obtain a generalization of the Two-Square Lemma proved for abelian categories by Fay, Hardie, and Hilton in 1989 and (in a special case) for preabelian categories by Generalov in 1994. We also prove the equivalence up to sign of two definitions of a connecting morphism of the Snake Lemma.