Koszul duality for operadic categories

TL;DR

This paper extends Koszul duality theory to a broad class of operadic categories, providing concrete examples and proving Koszulity for various operad types including classical, cyclic, modular, and PROP-like structures.

Contribution

It establishes the foundations of Koszul duality in operadic categories and demonstrates Koszulity for numerous operad variants and complex algebraic structures.

Findings

Proved Koszulity for classical and exotic operads

Described Koszul duals for binary quadratic operads

Extended Koszul duality framework to diverse operadic categories

Abstract

The aim of this sequel to arXiv:1812.02935 is to set up the cornerstones of Koszul duality and Koszulity in the context of operads over a large class of operadic categories. In particular, for these operadic categories we will study concrete examples of binary quadratic operads, describe their Koszul duals and prove that they are Koszul. This includes operads whose algebras are the most important operad- and PROP-like structures such as the classical operads, their variants such as cyclic, modular or wheeled operads, and also diverse versions of PROPs such as properads, dioperads, 1/2PROPs, and still more exotic objects such as permutads and pre-permutads.