In the context of geometric Langlands duality, it is a general principle that the "topological" aspects (e.g., intersection cohomology, perverse sheaves) of a given group G should correspond to "algebraic" aspects (e.g., representations, coherent sheaves) of its dual group G'. An archetypal instance of this idea is the "geometric Satake isomorphism" of Ginzburg and Mirkovic-Vilonen, but by now there are many results asserting an equivalence (often derived) between a topological category associated to G and an algebraic one associated to G'. In this talk, I will try to explain a few examples of this phenomenon, which can give rise to surprising kinds of objects: coherent sheaves (in characteristic 0) that behave as though they were perverse, or vice versa. I will also say a few words about the titular objects, which don't (yet?) come from such an equivalence.