Sheaves on Triangulated Spaces and Koszul Duality

Let $X$ be a finite connected simplicial complex, and let $\delta$ be a perversity (i.e., some function from integers to integers). One can consider two categories: (1) the category of perverse sheaves cohomologically constructible with respect to the triangulation, and (2) the category of sheaves constant along the perverse simplices ($\delta$-sheaves). We interpret the categories (1) and (2) as categories of modules over certain quadratic (and even Koszul) algebras $A(X,\delta)$ and $B(X,\delta)$ respectively, and we prove that $A(X,\delta)$ and $B(X,\delta)$ 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 equivalent to the category of $\delta$ sheaves. Finally, we study the relationship between the Koszul duality functor and the Verdier duality functor for simplicial sheaves and cosheaves.