Categorical Koszul duality

In this paper we establish Koszul duality between dg categories and a class of curved coalgebras, generalizing the corresponding result for algebras conilpotent coalgebras. We show that normalized chain complex functor transforms Quillen equivalence quasicategories simplicial into duality. This allows us to give conceptual interpretation nerve category its adjoint. As an application, prove representations quasicategory K is equivalent coderived comodules over C⁎(K), coalgebra K. A corollary characterization constructible sheaves on stratified space as certain coalgebra.