Stability Conditions on Abelian Comma Categories

TL;DR

This paper investigates the structural properties of abelian comma categories, providing conditions for their abelian, noetherian, and artinian nature, and introduces a framework for stability conditions within these categories.

Contribution

It establishes criteria for when comma categories are abelian, noetherian, and artinian, and develops a method to induce stability conditions between initial and comma categories.

Findings

Provided sufficient conditions for abelian structure

Computed the Grothendieck group of comma categories

Established criteria for noetherian and artinian properties

Abstract

A comma category, exemplified in algebraic geometry by coherent systems, combines two categories over a third through morphisms between their objects. We establish sufficient conditions for it to be abelian, compute its Grothendieck group, and give necessary and sufficient criteria for it to be noetherian and artinian. Finally, we define a stability condition on abelian comma categories under hypotheses on the initial categories and, conversely, induce stability conditions on the initial abelian categories from those on the comma categories.