Let X be a simplicial complex, and let delta be a perversity (i.e., some function from integers to integers). One can consider two categories: the category of perverse sheaves cohomologically constructible with respect to the triangulation, and the category of sheaves constant along the perverse simplices (delta-sheaves). We present perverse sheaves and delta-sheaves as modules over certain quadratic algebras, and prove that the categories of perverse sheaves and delta-sheaves are Koszul dual to each other. We define the delta-perverse topology on X and prove that the category of sheaves on perverse topology is also Koszul dual to the category of perverse sheaves. Finally, we study the relationship between the Koszul duality functor and the Verdier duality functor for simplicial sheaves and cosheaves.
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