Relative nonhomogeneous Koszul duality for PROPs related to nonaugmented operads
Authors: Geoffrey Powell
Summary: The aim of this paper is to point out how Positselski’s relative nonhomogeneous Koszul duality principle applies when learning the linear class underlying the PROP related to a (non-augmented) operad of a sure type, particularly assuming that the diminished a part of the operad is binary quadratic. On this case, the linear class has each a left augmentation and a proper augmentation (equivalent to completely different models), utilizing Positselski’s terminology. The overall principle offers two related linear differential graded (DG) classes; certainly, on this framework, one can work totally inside the DG realm, versus the curved setting required for Positselski’s common principle. Furthermore, DG modules over DG classes are associated by adjunctions. When the diminished a part of the operad is Koszul (working over a subject of attribute zero), the relative Koszul duality principle reveals that there’s a Koszul-type equivalence between the suitable homotopy classes of DG modules. This offers a type of Koszul duality relationship between the above DG classes. That 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. On this case, the related linear class is the linearization of the class of finite units and all maps. The relative nonhomogeneous Koszul duality principle relates its derived class to the respective homotopy classes of modules over two specific linear DG classes.