Categorification of Harder–Narasimhan theory via slope functions valued in totally ordered sets

We introduce a categorical construction of Harder–Narasimhan filtration via a slope method which does not need a degree function. With a theorem of existence and uniqueness of Harder–Narasimhan filtration in our categorical setting, we give a categorical interpretation of Stuhler–Grayson filtration in the case of not necessarily Hermitian normed lattices.