Recent developments in Categorical Galois theory
by
George Janelidze
Mathematical Institute of the Georgian Academy of Sciences (Tbilisi, Georgia) and Aveiro University (Aveiro, Portugal)

Categorical Galois theory (see [1], [3], [4], [7], and references there) generalizes A. Grothendieck’s extension of classical Galois theory (including the classification of covering maps in algebraic geometry and algebraic topology), A. R. Magid’s separable Galois theory of commutative rings, differential Galois theory, and the theories of central extensions in homological and universal algebra – and has some other important known and new examples. We briefly recall these 10-15 years old results, and then describe three directions of recent developments: second order coverings of simplicial sets [2], universal-algebraic theory of central extensions [5], [6], and Tannaka duality [8]. In addition we discuss the relationship between the classical viewpoint (“Galois connections”) and the categorical viewpoint (“adjoint functors”).

References

1. F. Borceux and G. Janelidze: Galois Theories, Cambridge University Press.

2. R. Brown and G. Janelidze: Galois theory of second order covering maps of simplicial sets, Journal of Pure and Applied Algebra 135, 1999, 23-31.

3. G. Janelidze: Pure Galois theory in categories, Journal of Algebra 132, 1990, 270-286.

4. G. Janelidze: Precategories and Galois theory, Springer Lecture Note in Mathematics 1488, 1991, 157-173.

5. G. Janelidze and G. M. Kelly: Central extensions in Mal’tsev varieties, Theory and Applications of categories 7, 10, 2000, 219-226.

6. G. Janelidze and G. M. Kelly: Central extensions in universal algebra: a unification of three notions, Algebra Universalis 44, 2000, 123-128.

7. G. Janelidze, D. Schumacher, R. H. Street: Galois theory in variable categories, Applied Categorical Structures 1, 1993, 103-110.

8. G. Janelidze and R. H. Street: Galois theory in symmetric monoidal categories, Journal of Algebra 220, 1999, 174-187.

Date received: January 22, 2001