Advertisement

Extracting a \(\Sigma \)-Mal’tsev (\(\Sigma \)-Protomodular) Structure from a Mal’tsev (Protomodular) Subcategory

  • Dominique BournEmail author
Article
  • 1 Downloads

Abstract

We give conditions on an inclusion \({\mathbb {C}}\hookrightarrow {\mathbb {D}}\) where \({\mathbb {C}}\) is a Mal’tsev (resp. protomodular) subcategory in order to produce on \({\mathbb {D}}\) a partial \(\Sigma \)-Mal’tsev (resp. \(\Sigma \)-protomodular) structure.

Keywords

Reflections and reg-epi reflections Mal’tsev varieties and categories Protomodular varieties and categories Associated partial structures 

Mathematics Subject Classification

08B05 08C05 18A20 18A32 18C05 18E10 20Jxx 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

References

  1. 1.
    Barr, M.: Exact Categories. LNM, vol. 236, pp. 1–120. Springer, Berlin (1971)CrossRefzbMATHGoogle Scholar
  2. 2.
    Berger, C., Bourn, D.: Central reflections and nilpotency in exact Mal’tsev categories. J. Homotopy Relat. Struct. 12, 765–835 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Borceux, F., Bourn, D.: Mal’cev, Protomodular, Homological and Semi-Abelian Categories. Mathematics and Its Applications, vol. 566, p. 479. Kluwer, London (2004)CrossRefzbMATHGoogle Scholar
  4. 4.
    Borceux, F., Janelidze, G.: Galois Theories. Studies in Advanced Mathematics, vol. 72, p. 341. Cambridge University Press, Cambridge (2001)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bourn, D.: The shift functor and the comprehensive factorization. Cahiers Top. et Géom. Diff. Catégriques 28, 197–226 (1987)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bourn, D.: Normalization equivalence, kernel equivalence and affine categories. LNM, vol. 1488, pp. 43–62. Springer, Berlin (1991)zbMATHGoogle Scholar
  7. 7.
    Bourn, D.: Mal’cev categories and fibration of pointed objects. Appl. Categ. Struct. 4, 307–327 (1996)CrossRefzbMATHGoogle Scholar
  8. 8.
    Bourn, D.: The denormalized \(3\times 3\) lemma. J. Pure Appl. Algebra 177, 113–129 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bourn, D.: Abelian groupoids and non-pointed additive categories. Theory Appl. Categ. 20(4), 48–73 (2008)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Bourn, D.: Partial Mal’tsevness and partial protomodularity. arXiv: 1507.02886v1
  11. 11.
    Bourn, D.: A structural aspect of the category of Quandles. J. Knot Theory Ramif. 24(12), 1550060 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bourn, D.: Partial linearity and partial natural Mal’tsevness. Theory Appl. Categ. 31, 418–443 (2016)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Bourn, D., Janelidze, G.: Characterization of protomodular varieties of universal algebra. Theory Appl. Categ. 11(6), 143–147 (2003)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Bourn, D., Martins-Ferreira, N., Montoli, A., Sobral, M.: Monoids and pointed S-protomodular categories. Homol. Homotopy Appl. 18, 151–172 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bourn, D., Montoli, A.: The \(3\times 3\) lemma in the \(\Sigma \)-Mal’tsev and \(\Sigma \)-protomodular settings. Homol. Homotopy Appl. 21, 305–332 (2019)CrossRefGoogle Scholar
  16. 16.
    Carboni, A., Janelidze, G., Kelly, G.M., Paré, B.: On localization and stabilization for factorizations systems. Appl. Categ. Struct. 5, 1–58 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Carboni, A., Lambek, J., Pedicchio, M.C.: Diagram chasing in Mal’cev categories. J. Pure Appl. Algebra 69, 271–284 (1990)CrossRefzbMATHGoogle Scholar
  18. 18.
    Carboni, A., Pedicchio, M.C., Pirovano, N.: Internal graphs and internal groupoids in Mal’cev categories. In: CMS conference proceedings, vol. 13, pp. 97–109 (1992)Google Scholar
  19. 19.
    Cassidy, C., Herbert, M., Kelly, G.M.: Reflective subcategories, localizations and factorization systems. J. Aust. Math. Soc. (Ser. A) 38, 287–329 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Even, V., Gran, M., Montoli, A.: A characterization of central extensions in the variety of quandles. Theory Appl. Categ. 31, 201–216 (2016)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Gran, M.: Central extensions and internal groupoids in Mal’tsev categories. J. Pure Appl. Algebra 155, 139–166 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Janelidze, G., Kelly, G.M.: Galois theory and a general notion of central extension. J. Pure Appl. Algebra 97, 135–161 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Johnstone, P.T.: Affine categories and naturally Mal’cev categories. J. Pure Appl. Algebra 61, 251–256 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Joyce, D.: A classifying invariant of knots, the knot quandle. J. Pure Appl. Algebra 23, 37–65 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Mal’cev, A.I.: On the general theory of algebraic systems. Mat. Sb. (N.S.) 35, 3–20 (1954)MathSciNetGoogle Scholar
  26. 26.
    Matveev, S.V.: Distributive groupoids in knot theory. Mat. Sb. (N.S.) 119(161 No.1), 78–88 (1982)MathSciNetGoogle Scholar
  27. 27.
    Smith, J.D.H.: Mal’cev Varieties. LNM, vol. 554. Springer, Berlin (1976)CrossRefzbMATHGoogle Scholar
  28. 28.
    Ursini, A.: On subtractive varieties. I. Algebra Universalis 31, 204–222 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques Pures et AppliquéesUniversité du LittoralCalaisFrance

Personalised recommendations