Categorification of Harder-Narasimhan Theory via slope functions

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.