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Normalizers and Split Extensions

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Abstract

We make explicit a larger structural phenomenon hidden behind the existence of normalizers in terms of existence of certain precartesian maps related to the kernel functor.

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References

  1. Barr, M.: Exact categories. Springer L. N. Math. 236, 1–120 (1971)

    Google Scholar 

  2. Borceux, F., Clementino, M.M.: Topological semi-abelian algebras. Adv. Math. 190, 425–453 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Borceux, F., Janelidze, G., Kelly, G.M.: On the representability of actions in a semi-abelian category. Theory Appl. Categ. 14, 244–286 (2005)

    MATH  MathSciNet  Google Scholar 

  4. Bourn, D.: Normalization Equivalence, Kernel Equivalence and Affine Categories. Springer L. N. Math. 1488, 43–62 (1991)

    MathSciNet  Google Scholar 

  5. Bourn, D.: Mal’cev categories and fibration of pointed objects. Appl. Categ. Struct. 4, 43–62 (1996)

    Article  MathSciNet  Google Scholar 

  6. Bourn, D.: Normal subobjects and abelian objects in protomodular categories. J. Algebra 228, 143–164 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bourn, D.: Normal subobjects of topological groups and of topological semi-Abelian algebras. Topol. Appl. 153, 1341–1364 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bourn, D.: Two ways to centralizers of equivalence relations. Applied Categorical Structures, accepted and online (2013). doi:10.1007/s10485-013-9347-2

  9. Bourn, D., Borceux, F.: Mal’cev, protomodular, homological and semi-abelian categories Kluwer. Math. Appl. 566, 479 (2004)

    MathSciNet  Google Scholar 

  10. Bourn, D., Gran, M.: Centrality and normality in protomodular categories. Theory Appl. Categ. 9, 151–165 (2002)

    MathSciNet  Google Scholar 

  11. Bourn, D., Gray, J.R.A.: Aspects of algebraic exponentiation. Bull. Belg. Math. Soc. Simon Stevin 19, 823–846 (2012)

    MathSciNet  Google Scholar 

  12. Bourn, D., Janelidze, G.: Centralizers in action accessible categories. Cahiers de Top. et Géom. Diff. Catégoriques 50, 211–232 (2009)

    MATH  MathSciNet  Google Scholar 

  13. Bourn, D., Janelidze, Z.: Categorical (binary) difference terms and protomodularity Algebra Universalis. Algebra Univers. 66, 277–316 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. Carboni, A., Kelly, G.M.: Some remarks on maltsev and goursat categories. Appl. Categ. Struct. 1, 365–421 (1993)

    MathSciNet  Google Scholar 

  15. Carboni, A., Lambek, J., Pedicchio, M.C.: Diagram chasing in Mal’cev categories. J. Pure Appl. Algebra 69, 271–284 (1991)

    Article  MathSciNet  Google Scholar 

  16. Carboni, A., Pedicchio, M.C., Pirovano, N.: Internal graphs and internal groupoids in Mal’cev categories. CMS Conf. Proc. 13, 97–109 (1992)

    MathSciNet  Google Scholar 

  17. Cigoli, A., Mantovani, S.: Action accessibility and centralizers. J. Pure Appl. Algebra 216, 1852–1865 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  18. Gray, J.R.A.: Algebraic exponentiation and internal homology in general categories. PhD thesis, University of Cape To (2010)

  19. Gray, J.R.A.: Algebraic exponentiation in general categories. Appl. Categ. Struct. 20, 543–567 (2012)

    Article  MATH  Google Scholar 

  20. Gray, J.R.A.: Normalizers, centralizers and action representability in semi-abelian categories. Applied Categorical Structures, accepted and online (2014). doi:10.1007/s10485-014-9379-2

  21. Johnstone, P.T.: Sketches of an Elephant: A Topos Theory Compendium. Oxford University Press, Oxford (2002)

    Google Scholar 

  22. Lang, S.: Algebra. Addison-Wesley Publishing Company (1965)

  23. Pedicchio, M.C.: A categorical approach to commutator theory. J. Algebra 177, 143–147 (1995)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Laboratoire de Mathématiques Pures et Appliquées, Université du Littoral, Bat. H. Poincaré, 50 Rue F. Buisson, BP 699, 62228, Calais Cedex, France

    Dominique Bourn

  2. University of South Africa, Pretoria, South Africa

    James Richard Andrew Gray

Authors
  1. Dominique Bourn
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  2. James Richard Andrew Gray
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Correspondence to James Richard Andrew Gray.

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Bourn, D., Gray, J.R.A. Normalizers and Split Extensions. Appl Categor Struct 23, 753–776 (2015). https://doi.org/10.1007/s10485-014-9382-7

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  • DOI: https://doi.org/10.1007/s10485-014-9382-7

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