There are three equivalent exotic derived categories associated to a CDG-coalgebra C and the CDG-algebra Cob(C) cobar-dual to C: the coderived category of CDG-comodules over C, the contraderived category of CDG-contramodules over C, and the absolute derived category of CDG-modules over Cob(C). When C is coaugmented and conilpotent, Cob(C) is a DG-algebra whose absolute derived category coincides with its conventinal derived category; in this conilpotent case, one can formulate a more general triality theorem claiming the equivalence between the coderived category of CDG-comodules over C, the contraderived category of CDG-contramodules over C, and the derived category of DG-modules over A, for any CDG-coalgebra C and DG-algebra A for which an acyclic twisting cochain \tau\in Hom_k^1(C,A) is given.
Under the latter equivalences of categories, the trivial CDG-comodule k over C corresponds to the free DG-module A over A, while the trivial CDG-contramodule k over C corresponds to the cofee DG-module Hom_k(A,k) over A. When C is actually a DG-coalgebra and A is accordingly an augmented DG-algebra, one can also say that the cofree DG-comodule C over C corresponds to the free DG-contramodule Hom_k(C,k) over C and to the trivial DG-module k over A under these equivalences of categories.
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