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Galois Connections and Applications pp 139–171Cite as

Categorical Galois Theory: Revision and Some Recent Developments

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Part of the book series: Mathematics and Its Applications ((MAIA,volume 565))

Abstract

This expository paper presents a short review of categorical Galois theory, with special attention to the connection with A. R. Magid’s Galois theory of commutative rings and most recent developments in the theory of generalized central extensions. In the last section some open questions are proposed.

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References

  1. M. Auslander and O. Goldman, The Brauer group of a commutative ring, Trans. AMS 97, 1960, 367–409.

    Article  MathSciNet  Google Scholar 

  2. M. Barr, Abstract Galois theory, Journal of Pure and Applied Algebra 19, 1980, 21–42.

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Barr, Abstract Galois theory II, Journal of Pure and Applied Algebra 25, 1982, 227–247.

    Article  MathSciNet  MATH  Google Scholar 

  4. F. Borceux and G. Janelidze, Galois Theories, Cambridge Studies in Advanced Mathematics 72, Cambridge University Press, 2001.

    Book  Google Scholar 

  5. D. Bourn, Normalization equivalence, kernel equivalence and affine categories, Lecture Notes in Math. 1488, Springer, 1991, 43–62.

    Google Scholar 

  6. D. Bourn and M. Gran, Central extensions in semi-abelian categories, Journal of Pure and Applied Algebra, 175, 2002, 31–44.

    Article  MathSciNet  MATH  Google Scholar 

  7. D. Bourn, M. Gran, G. Janelidze and M. C. Pedicchio, On two commutators in semi-abelian categories, in preparation.

    Google Scholar 

  8. 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.

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Carboni and G. Janelidze, Decidable (separable) objects and morphisms in 1-extensive categories, Journal of Pure and Applied Algebra 110, 1996, 219–240.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Carboni and G. Janelidze, Boolean Galois theories, Georgian Mathematical Journal 9, 4, 2002, 645 - 658.

    MathSciNet  MATH  Google Scholar 

  11. A. Carboni, G. Janelidze, G. M. Kelly and R. Pare, On localization and stabilization of factorization system, Applied Categorical Structures 5, 1997, 1–58.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Carboni, G. Janelidze and A. R. Magid, A note on Galois correspondence for commutative rings, Journal of Algebra 183, 1996, 266–272.

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Carboni, G. M. Kelly and M. C. Pedicchio, Some remarks on Mal’tsev and Goursat categories, applied Categorical Structures 1, 1993, 385–421.

    MATH  Google Scholar 

  14. A. Carboni, J. Lambek and M. C. Pedicchio, Diagram chasing in Mal’tsev categories, Journal of Pure and Applied Algebra 69, 1990, 271–284.

    Article  MathSciNet  MATH  Google Scholar 

  15. S. U. Chase, D. K. Harrison and A. Rosenberg, Galois theory and cohomology of commutative rings, Mem. AMS 52, 1965, 15–33.

    MathSciNet  MATH  Google Scholar 

  16. S. U. Chase and M. E. Sweedler, Hopf algebras and Galois theory, Lecture Notes in Math. 97, Springer 1969.

    Google Scholar 

  17. P. Deligne, Categories tannakiennes,“Grothendieck Festschrift” 2, Birkhäuser 1990, 111–195.

    Google Scholar 

  18. F. R. DeMeyer and E. Ingraham, Separable algebras over a commutative ring, Lecture Notes in Math. 181, Springer 1971.

    Google Scholar 

  19. Y. Diers, Categories of Boolean sheaves of simple algebras, Lecture Notes in Math. 1187, Springer 1986.

    Google Scholar 

  20. A. Fröhlich, Baer-invariants of algebras, Trans. AMS 109, 1963, 221–244.

    Article  MATH  Google Scholar 

  21. J. Furtado-Coelho, Varieties of Ω -groups and associated functors, Ph. D. Thesis, University of London, 1972.

    Google Scholar 

  22. M. Gran, Algebraically central and categorically central extensions, Coimbra University Math. Preprint 01–02, 2001.

    Google Scholar 

  23. M. Gran, Commutators and central extension in universal algebra, Journal of Pure and Applied Algebra 174, 2002, 249 -261.

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Grothendieck, Revêtements étale et groupe fundamental, SGA 1, expose V, Lecture Notes in Math. 224, Springer 1971.

    Google Scholar 

  25. D. K. Harrison, Abelian extensions to commutative rings, Mem. AMS 52, 1965, 1–14.

    MATH  Google Scholar 

  26. S.A. Huq, Commutator, nilpotency and solvability in categories, Quart. Journal Math. Oxford (2) 19, 1968, 363–389.

    Article  MathSciNet  MATH  Google Scholar 

  27. G. Janelidze, Magid’s theorem in categories, Bull. Georgian Acad. Sei. 114, 3, 1984, 497–500 (in Russian).

    Article  MathSciNet  Google Scholar 

  28. G. Janelidze, The fundamental theorem of Galois theory, Math. USSR Sbornik 64 (2), 1989, 359–384.

    Article  MathSciNet  Google Scholar 

  29. G. Janelidze, Galois theory in categories: the new example of differential fields, Proc. Conf. Categorical Topology in Prague 1988, World Scientific 1989, 369–380.

    Google Scholar 

  30. G. Janelidze, Pure Galois theory in categories, Journal of Algebra 132, 1990, 270–286.

    Article  MathSciNet  MATH  Google Scholar 

  31. G. Janelidze, What is a double central extension? (the question was asked by Ronald Brown), Cahiers de Topologie et Geometrie Différentielle Catégorique XXXII-3, 1991, 191–202.

    Google Scholar 

  32. G. Janelidze, Precategories and Galois theory, Lecture Notes in Math. 1488, Springer, 1991, 157–173.

    Google Scholar 

  33. G. Janelidze, Higher Dimensional Central Extensions: A Categorical Approach to Homology Theory of Groups, Talk at the International Meeting in Category Theory, Halifax (Canada), 1995.

    Google Scholar 

  34. G. Janelidze and G. M. Kelly, Galois theory and a general notion of a central extension, Journal of Pure and Applied Algebra 97, 1994, 135–161.

    Article  MathSciNet  MATH  Google Scholar 

  35. G. Janelidze and G. M. Kelly, The reflectiveness of covering morphisms in algebra and geometry, Theory and Applications of Categories 3, 1997, 132–159.

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  37. G. Janelidze and G. M. Kelly, Central extensions in Mal’tsev varieties, Theory and Application of Categories 7, 10, 219–226.

    MathSciNet  Google Scholar 

  38. G. Janelidze, L. Marki and W. Tholen, Locally semisimple coverings, Journal of Pure and Applied Algebra 128, 1998, 281–289.

    Article  MathSciNet  MATH  Google Scholar 

  39. G. Janelidze, L. Marki and W. Tholen, Semi-abelian categories, Journal of Pure and Applied Algebra, 168, 2002, 367 - 386.

    Article  MathSciNet  MATH  Google Scholar 

  40. G. Janelidze and M. C. Pedicchio, Pseudogroupoids and commutators, Theory and Applications of Categories 8, 15, 2001, 408 - 456.

    MathSciNet  MATH  Google Scholar 

  41. G. Janelidze, D. Schumacher and R. H. Street, Galois theory in variable categories, Applied Categorical Structures 1, 1993, 103–110.

    Article  MathSciNet  MATH  Google Scholar 

  42. G. Janelidze and M. Sobral, Finite preorders and topological descent I, Journal of Pure and Applied Algebra, 175 (1–3), 2002, 187 - 205.

    Article  MathSciNet  MATH  Google Scholar 

  43. G. Janelidze and M. Sobral, Finite preorders and topological descent II: Etale descent, Journal of Pure and Applied Algebra, 174, 2002, 303 - 309.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  45. G. Janelidze and W. Tholen, Facets of Descent I, Applied Categorical Structures 2, 1994, 245–281.

    Article  MathSciNet  MATH  Google Scholar 

  46. G. Janelidze and W. Tholen, Facets of Descent II, Applied Categorical Structures 5, 1997, 229–248.

    Article  MathSciNet  MATH  Google Scholar 

  47. G. Janelidze and W. Tholen, Functorial factorization, well-pointedness and separability, Journal of Pure and Applied Algebra 142, 1999, 99–130.

    Article  MathSciNet  MATH  Google Scholar 

  48. G. Janelidze and W. Tholen, Extended Galois theory and dissonant morphisms, Journal of Pure and Applied Algebra 143, 1999, 231–253.

    Article  MathSciNet  MATH  Google Scholar 

  49. G. Janelidze and W. Tholen, Strongly separable morphisms in general categories, in preparation.

    Google Scholar 

  50. G. Janelidze and W. Tholen, Facets of Descent III, in preparation.

    Google Scholar 

  51. G. J. Janusz, Separable algebras over commutative rings, Trans. AMS 122, 1966, 461–479.

    Article  MathSciNet  MATH  Google Scholar 

  52. A. Joyal and M. Tierney, An extension of the Galois theory of Grothendieck, Mem. AMS, 1984.

    Google Scholar 

  53. A.S.-T. Lue, Bear-invariants and extensions relative to a variety, Proc. Cambridge Philos. Soc. 63, 1967.

    Google Scholar 

  54. S. Mac Lane, Galois theory in categories (work of G. Janelidze), Talk on International Meeting in Category Theory, Lovain-la-Neuve (Belgium), 1987.

    Google Scholar 

  55. A. R. Magid, The separable Galois theory of commutative rings, Marcel Dekker, 1974.

    MATH  Google Scholar 

  56. A. R. Magid, Review on “ Selected works of Ellis Kolchin”, Bulletin AMS 37, 3, 2000, 337–342.

    Article  Google Scholar 

  57. L. Marki, R. Mlitz and R. Wiegandt, A general Kurosh-Amitsur radical theory, Communications in Algebra 16, 1988, 249–305.

    Article  MathSciNet  MATH  Google Scholar 

  58. M. C. Pedicchio, A categorical approach to commutator theory, Journal of Algebra 177, 1995, 647–657.

    Article  MathSciNet  MATH  Google Scholar 

  59. R. S. Pierce, Modules over commutative regular rings, Mem. AMS 70, 1967.

    Google Scholar 

  60. D. Quillen, Homotopical algebra, Lecture Notes in Math. 43, Springer, 1967.

    Book  Google Scholar 

  61. J. Reiterman and W. Tholen, Effective descent maps of topological spaces, Topology and its Applications 57, 1994, 53–69.

    Article  MathSciNet  MATH  Google Scholar 

  62. J. D. H. Smith, Mal’tsev varieties, Lecture Notes in Math. 554, Springer, 1976.

    Google Scholar 

  63. O. Villamayor and D. Zelinsky, Galois theory for rings with finitely many idempotents, Nagoya Math. Journal 27, 1966, 721–731.

    MathSciNet  Google Scholar 

  64. O. Villamayor and D. Zelinsky, Galois theory with infinitely many idempotents, Nagoya Math. Journal 35, 1969, 83–98.

    MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. A. Razmadze Mathematical Institute, Georgian Academy of Sciences, 1 M. Alexidze Street, Tbilisi, 0193, Georgia

    G. Janelidze

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  1. G. Janelidze
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Editor information

Editors and Affiliations

  1. University of Potsdam, Potsdam, Germany

    K. Denecke

  2. University of Hannover, Hannover, Germany

    M. Erné

  3. University of Lethbridge, Lethbridge, Canada

    S. L. Wismath

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Janelidze, G. (2004). Categorical Galois Theory: Revision and Some Recent Developments. In: Denecke, K., Erné, M., Wismath, S.L. (eds) Galois Connections and Applications. Mathematics and Its Applications, vol 565. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-1898-5_2

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  • DOI: https://doi.org/10.1007/978-1-4020-1898-5_2

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-6540-7

  • Online ISBN: 978-1-4020-1898-5

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