Non-Abelian Homological Algebra and Its Applications

This text exposes methods of non-abelian homological algebra, such as the theory of satellites in abstract categories with respect to presheaves of categories and the theory of non-abelian derived functors of group valued functors.

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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.