An algebraic approach to Harder-Narasimhan filtrations

In this article we study chains of torsion classes in an abelian category $\mathcal{A}$. We prove that each chain of torsion classes induce a Harder-Narasimhan filtration for every nonzero object $M$ in $\mathcal{A}$, generalising a well-known property of stability conditions. We also characterise the slicings of $\mathcal{A}$ in terms of chain of torsion classes. We finish the paper by showing that chains of torsion classes induce wall-crossing formulas in the completed Hall algebra of the category.