Categorification of Harder-Narasimhan Theory via slope functions

TL;DR

This paper develops a categorical approach to Harder-Narasimhan filtration using slope functions, removing the need for additive degree functions and sub-quotient structures, thus broadening its applicability.

Contribution

It introduces a new categorical construction of Harder-Narasimhan filtration that does not rely on additive degree functions or sub-quotient structures.

Findings

Provides a method for proving existence and uniqueness of filtrations

Generalizes Harder-Narasimhan theory beyond vector bundles

Removes the need for additive degree functions in the construction

Abstract

The notion of Harder-Narasimhan filtration was firstly introduced by Harder and Narasimhan in the setting of vector bundles on a non-singular projective curve. Curiously, analogous constructions have been discovered in other branches of mathematics which motivate categorical constructions of Harder-Narasimhan filtration. In this article, we introduce a categorical construction of Harder-Narasimhan filtration via slope function method which does not need the additive condition in degree function. We also give a method to prove the existence and uniqueness theorem of the Harder-Narasimhan filtration intrinsically from a set E and a admissible collection of subsets of E, which does not need sub-quotient structure.