Many familiar objects in algebra, geometry and topology belong to 2-Calabi-Yau categories, e.g. coherent sheaves on a K3 or Abelian surface, Higgs bundles on a smooth and proper curve, local systems on a Riemann surface, and representations of a fixed preprojective algebra. In each case the Borel-Moore homology of the stack of objects carries a Hall algebra structure, which in the preprojective algebra case contains all of the raising operators on the cohomology of Nakajima quiver varieties, and in other cases produces a kind of nonlinear/global analogue of this theory. By reducing everything locally to the case of representations of a preprojective algebra, I'll show that in each case the Borel-Moore homology of the stack of objects carries a perverse filtration. I'll explain how this filtration can be used to produce generators for the Borel-Moore homology of the stack of objects coming from intersection cohomology of the coarse moduli space, along with tautological classes. In joint work with Sjoerd Beentjes, we show that in all but one of the examples in the first sentence of this abstract this Borel-Moore homology is pure. In the K3 case this proves a conjecture of Halpern-Leistner.