One-sided exact categories

TL;DR

This paper explores one-sided exact categories, their properties, and their relation to homological lemmas and derived categories, extending classical results to this new categorical framework.

Contribution

It introduces and studies one-sided exact additive categories, including a stronger version with additional axioms, and demonstrates their homological properties and derived category construction.

Findings

Homological lemmas like Short Five Lemma hold in this setting

Derived categories can be constructed for one-sided exact additive categories

Connections to Grothendieck pretopologies and Quillen axioms are established

Abstract

One-sided exact categories appear naturally as instances of Grothendieck pretopologies. In an additive setting they are given by considering the one-sided part of Keller's axioms defining Quillen exact categories. We study one-sided exact additive categories and a stronger version defined by adding the one-sided part of Quillen "obscure axiom". We show that some homological results, such as the Short Five Lemma and the 3 X 3 Lemma, can be proved in our context. We also note that the derived category of a one-sided exact additive category can be constructed.