Relative nonhomogeneous Koszul duality for PROPs associated to nonaugmented operads

TL;DR

This paper extends Positselski's relative nonhomogeneous Koszul duality to the linear categories associated with certain non-augmented operads, especially those with binary quadratic reduced parts, establishing dualities in the DG setting.

Contribution

It applies relative Koszul duality to the linear categories of non-augmented operads, providing explicit DG category dualities and extending classical results to new operadic contexts.

Findings

Establishes Koszul-type equivalences between homotopy categories of DG modules.

Extends classical Koszul duality to operads encoding unital, commutative associative algebras.

Relates derived categories of linear categories to homotopy categories of modules over explicit DG categories.

Abstract

The purpose of this paper is to show how Positselski's relative nonhomogeneous Koszul duality theory applies when studying the linear category underlying the PROP associated to a (non-augmented) operad of a certain form, in particular assuming that the reduced part of the operad is binary quadratic. In this case, the linear category has both a left augmentation and a right augmentation (corresponding to different units), using Positselski's terminology. The general theory provides two associated linear differential graded (DG) categories; indeed, in this framework, one can work entirely within the DG realm, as opposed to the curved setting required for Positselski's general theory. Moreover, DG modules over DG categories are related by adjunctions. When the reduced part of the operad is Koszul (working over a field of characteristic zero), the relative Koszul duality theory shows that there is a Koszul-type equivalence between the appropriate homotopy categories of DG modules. This gives a form of Koszul duality relationship between the above DG categories. This is illustrated by the case of the operad encoding unital, commutative associative algebras, extending the classical Koszul duality between commutative associative algebras and Lie algebras. In this case, the associated linear category is the linearization of the category of finite sets and all maps. The relative nonhomogeneous Koszul duality theory relates its derived category to the respective homotopy categories of modules over two explicit linear DG categories.