Homological and Homotopical Algebra and Category Theory

Dear Colleagues,

The foundations of homological and homotopical algebra go back to the latter half of the previous century, to the works of S. Eilenberg, S. MacLane, and H. Cartan, and further to follow A. Dold and D. Quillen. Its aim is to study the properties of algebraic objects (groups, associative rings and algebras, Lie algebras, modules, monoids) using the methods of algebraic topology. It can also be viewed as a linearized version of homotopy theory of homotopy types. Nowadays, homological and homotopical algebra is a profound branch of mathematics, a fundamental and essential tool useful for many areas of mathematics, for example, class field theory, algebraic topology, and homotopy theory. That is illustrated by the famous Serre’s Conjecture, regarding the relationship between projective modules and free modules over polynomial rings. This problem of homological and commutative algebra, coming from algebraic geometry, was affirmatively proven in 1976. It has also helped to rewrite the foundations of algebraic geometry, to prove Weil Conjectures, and to create very powerful areas, such as homology theory of groups, Hochschild and cyclic homology theories, and algebraic K-theory. It should be noted that K-theory (algebraic and topological), based on fundamental works of A. Grothendieck, M. Atiyah, H. Bass, J. Milnor, and D.Quillen, is closely related to homotopical algebra and non-commutative geometry having applications to theoretical physics (string theory).

The beginnings of category theory go to A. Grothendieck and F.W. Lawvere in the 20th century. Category theory has come to occupy a central position in contemporary mathematics and theoretical computer science. It is a general mathematical theory formalizing mathematical structures and their concept in terms of objects and arrows called morphisms. It is a powerful language, allowing us to see the universal components of a family of structures of a given kind, and how mathematical structures of different kinds are interrelated. It can be applied to mathematical logic and it is an alternative to set theory as a foundation of mathematics. Categorical methods are successfully used in homological algebra and  algebraic topology.

Prof. Dr. Hvedri Inassaridze
Prof. Dr. Antonio Cegarra
Prof. Dr. Marino Gran
Prof. Dr. Michael Batanin
Prof. Dr. Zurab Janelidze
Guest Editors

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