Harder-Narasimhan categories

TL;DR

This paper introduces arithmetic exact categories, generalizing Quillen's exact categories, and develops a framework for Harder-Narasimhan filtrations and polygons with functorial properties, extending classical formalism.

Contribution

It proposes a new class of categories called arithmetic exact categories and establishes conditions for defining Harder-Narasimhan filtrations and polygons within them, including their functoriality.

Findings

Defined arithmetic exact categories as a generalization of Quillen's exact categories.

Established conditions for Harder-Narasimhan filtrations and polygons in these categories.

Proved the functoriality of Harder-Narasimhan filtrations indexed by real numbers.

Abstract

We propose a generalization of Quillen's exact category -- arithmetic exact category and we discuss conditions on such categories under which one can establish the notion of Harder-Narasimhan filtrations and Harder-Narsimhan polygons. Furthermore, we show the functoriality of Harder-Narasimhan filtrations (indexed by $\mathbb R$), which can not be stated in the classical setting of Harder and Narasimhan's formalism.