Nonabelian Hodge theory and the decomposition theorem for 2-CY categories

Examples of 2CY categories include the category of coherent sheaves on a K3 surface, the category of Higgs bundles, and the category of modules over preprojective algebras or fundamental group algebras of compact Riemann surfaces.  Let p:M->N be the morphism from the stack of semistable objects in a 2CY category to the coarse moduli space.  I'll explain, using cohomological DT theory, formality in 2CY categories, and structure theorems for good moduli stacks, how to prove a version of the BBDG decomposition theorem for the exceptional direct image of the constant sheaf along p, even though none of the usual conditions for the decomposition theorem apply: p isn't projective or representable, M isn't smooth, the constant mixed Hodge module complex Q_M isn't pure...  As an application, I'll explain how this allows us to extend nonabelian Hodge theory to Betti/Dolbeault stacks.