While in classical (abelian) homological algebra additive functors from
abelian (or additive) categories to abelian categories are investigated,
non- abelian homological algebra deals with non-additive functors and
their homological properties, in particular with functors having values
in non-abelian categories. Such functors haveimportant applications in
algebra, algebraic topology, functional analysis, algebraic geometry and
other principal areas of mathematics. To study homological properties of
non-additive functors it is necessary to define and investigate their
derived functors and satellites. It will be the aim of this book based
on the results of researchers of A. Razmadze Mathematical Institute of
the Georgian Academy of Sciences devoted to non-abelian homological
algebra. The most important considered cases will be functors from
arbitrary categories to the category of modules, group valued functors
and commutative semigroup valued functors. In Chapter I universal
sequences of functors are defined and in- vestigated with respect to
(co)presheaves of categories, extending in a natural way the satellites
of additive functors to the non-additive case and generalizing the
classical relative homological algebra in additive categories to
arbitrary categories. Applications are given in the furth- coming
chapters. Chapter II is devoted to the non-abelian derived functors of
group valued functors with respect to projective classes using
projective pseu- dosimplicial resolutions. Their functorial properties
(exactness, Milnor exact sequence, relationship with cotriple derived
functors, satellites and Grothendieck cohomology, spectral sequence of
an epimorphism, degree of an arbitrary functor) are established and
applications to ho- mology and cohomology of groups are given.
