The aim of this paper is to construct the derived nonhomogeneous Koszul duality. The author considers the derived categories of DG-modules, DG-comodules, and DG-contramodules, the coderived and contraderived categories of CDG-modules, the coderived category of CDG-comodules, and the contraderived category of CDG-contramodules. The equivalence between the latter two categories (the comodule-contramodule correspondence) is established. Nonhomogeneous Koszul duality or "triality" (an equivalence between exotic derived categories corresponding to Koszul dual (C)DG-algebra and CDG-coalgebra) is obtained in the conilpotent and nonconilpotent versions. Various A-infinity structures are considered, and a number of model category structures are described. Homogeneous Koszul duality and D-$\Omega$ duality are discussed in the appendices.

Table of Contents

• Introduction
• Derived category of DG-modules
• Derived categories of DG-comodules and DG-contramodules
• Coderived and contraderived categories of CDG-modules
• Coderived category of CDG-comodules and contraderived category of CDG-contramodules
• Comodule-contramodule correspondence
• Koszul duality: Conilpotent and nonconilpotent cases
• $\mathrm{A}_\infty$-algebras and curved $\mathrm{A}_\infty$-coalgebras
• Model categories of DG-modules, CDG-comodules, and CDG-contramodules
• Model categories of DG-algebras and CDG-coalgebras
• Appendix A. Homogeneous Koszul duality
• Appendix B. $\mathcal{D}$-$\Omega$ duality
• Bibliography