Differential graded Koszul duality: an introductory survey

TL;DR

This paper provides a pedagogical overview of derived nonhomogeneous Koszul duality over a field, focusing on the duality between DG-algebras and curved DG-coalgebras, and the related module and comodule structures.

Contribution

It offers an introductory survey summarizing the covariant duality and triality structures in differential graded algebra and coalgebra theory, based on the author's previous work.

Findings

Clarifies the duality between DG-algebras and curved DG-coalgebras

Explains the triality among DG-modules, CDG-comodules, and CDG-contramodules

Provides historical context and personal insights

Abstract

This is an overview on derived nonhomogeneous Koszul duality over a field, mostly based on the author's memoir arXiv:0905.2621. The paper is intended to serve as a pedagogical introduction and a summary of the covariant duality between DG-algebras and curved DG-coalgebras, as well as the triality between DG-modules, CDG-comodules, and CDG-contramodules. Some personal reminiscences are included as a part of historical discussion.