I will talk about a joint project with Chenjing Bu, Ben Davison, Andrés Ibáñez-Núñez, and Tasuki Kinko. For a large class of stacks, we decompose their cohomology in terms of what we call BPS cohomology, which is a structure originating in enumerative geometry of Calabi-Yau 3-folds, but which is of interest beyond this class of examples. Such stacks include smooth stacks (such as the moduli of G-bundles on a curve), symplectic stacks (such as the moduli of G-Higgs bundle on a curve), or some (-1)-shifted symplectic stacks (such as the moduli of semistable sheaves on a Calabi-Yau threefold).