Koszul duality for categories with a fixed object set

TL;DR

This paper generalizes Koszul duality to categories with fixed objects, defining a duality for monoid objects in biclosed monoidal categories, and explores its implications for enriched categories and operads.

Contribution

It introduces a broad framework for Koszul duality in monoidal biclosed categories, extending classical algebraic notions to enriched categories and operads.

Findings

Defined Koszul dual of monoid objects in biclosed categories

Established adjunctions between module categories over C and K(C)

Connected Koszul duality of operads to the broader categorical context

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.