2 LEONID POSITSELSKI
ways. It is both the quotient category of the homotopy category by the thick
subcategory of complexes with zero cohomology and the triangulated subcategory
of the homotopy category formed by the complexes of projective or injective objects.
In the general case, this simple picture splits in two halves. The derived category of
the first kind is still defined as the quotient category of the homotopy category by
the thick subcategory of complexes (DG-modules, . . . ) with zero cohomology. It can
be also obtained as a full subcategory of the homotopy category, but the description
of this subcategory is more complicated [50, 29, 9]. On the other hand, the derived
category of the second kind is defined as the quotient category of the homotopy
category by a thick subcategory with a rather complicated description. At the
same time, it is equivalent to the full subcategory of the homotopy category formed
by complexes (DG-comodules, DG-contramodules, . . . ) which become injective or
projective when considered without the differential.
0.3. The time has come to mention that there exist two kinds of module cat-
egories for a coalgebra: besides the familiar comodules, there are also contramod-
ules . Comodules can be thought of as discrete modules which are unions of
their finite-dimensional subcomodules, while contramodules are modules where cer-
tain infinite summation operations are defined. For example, the space of linear
maps from a comodule to any vector space has a natural contramodule structure.
The derived category of the first kind is what is known as just the derived
category: the unbounded derived category, the derived category of DG-modules, etc.
The derived category of the second kind comes in two dual versions: the coderived
and the contraderived category. The coderived category works well for comodules,
while the contraderived category is useful for contramodules. The classical notion
of a DG-(co)algebra itself can be generalized in two ways; the derived category of
the first kind is well-defined for an A∞-algebra, while the derived category of the
second kind makes perfect sense for a CDG-coalgebra.
Other situations exist when derived categories of the second kind are well-
behaved. One of them is that of a CDG-ring whose underlying graded ring has
a finite homological dimension. In this case, the coderived and contraderived cat-
egories coincide. In particular, this includes the case of a CDG-algebra whose
underlying graded algebra is free. Such CDG-algebras can be thought of as strictly
counital curved A∞-coalgebras; CDG-modules over the former with free and cofree
underlying graded modules correspond to strictly counital curved A∞-comodules
and A∞-contramodules over the latter. For a cofibrant associative DG-algebra, the
derived, coderived, and contraderived categories of DG-modules coincide. Since
any DG-algebra is quasi-isomorphic to a cofibrant one, it follows that the derived
category of DG-modules over any DG-algebra can be also presented as a coderived
and contraderived category.
The functors of forgetting the differentials, assigning graded (co/contra)mod-
ules to CDG-(co/contra)modules, play a crucial role in the whole theory of de-
rived categories of the second kind. So it is helpful to have versions of these
functors defined for arbitrary DG-categories. An attempt to obtain such forget-
ful functors leads to a nice construction of an almost involution on the category
of DG-categories. The related constructions for CDG-rings and CDG-coalgebras
are important for the nonhomogeneous quadratic duality theory, particularly in the
relative case .