Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence

This paper can be thought of as an extended introduction to arXiv:0708.3398; nevertheless, most of its results are not covered by loc. cit. We consider the derived categories of DG-modules, DG-comodules, and DG-contramodules, the coderived and contraderived categories of CDG-modules, the coderived categories of CDG-comodules, and the contraderived categories 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 ©DG-algebra and CDG-coalgebra) is obtained in the conilpotent and nonconilpotent versions. Various $A_{\mathrm{\infty }}$-structures are considered, and a number of model category structures are described. Homogeneous Koszul duality and $D$-$\mathrm{\Omega }$ duality are discussed in the appendices.