Koszul duality for categories with a fixed object set

Abstract

We define a notion of Koszul dual of a monoid object in a monoidal biclosed model category. Our construction generalizes the classic Yoneda algebra Ext A (k, k). We apply this general construction to define the Koszul dual of a category enriched over spectra or chain complexes. This example relies on the classical observation that enriched categories are monoid objects in a category of enriched graphs. We observe that the category of enriched graphs is biclosed, meaning that it comes with both left and right internal hom objects. Given a category R (which plays the role of the ground field k in classical algebra), and an augmented R-algebra C, we define the Koszul dual of C, denoted K(C), as the R-algebra of derived endomorphisms of R in the category of right C-modules.We establish the expected adjunctions between the categories of modules over C and modules over K(C). We investigate the question of when the map from C to its double dual K(K(C)) is an equivalence. We also show that Koszul duality of operads can be understood as a special case of Koszul duality of categories. In this way we incorporate Koszul duality of operads in a wider context, and possibly clarify some aspects of it.