Groups up to congruence relation and from categorical groups to c-crossed modules

Abstract

We introduce a notion of c-group, which is a group up to congruence relation and consider the corresponding category. Extensions, actions and crossed modules (c-crossed modules) are defined in this category and the semi-direct product is constructed. We prove that each categorical group gives rise to a c-group and to a c-crossed module, which is a connected, special and strict c-crossed module in the sense defined by us. The results obtained here will be applied in the proof of an equivalence of the categories of categorical groups and connected, special and strict c-crossed modules.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Baez, J.C., Lauda, A.D.: Higher-dimensional algebra. V. 2-groups. Theory Appl. Categ. 12, 423–491 (2004)

  2. 2.

    Breen, L.: Théorie de Schreier supérieure. Ann. Sci. École Norm. Sup. 25, 465–514 (1992)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Brown, R., Higgins, P.J., Sivera, R.: Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy groupoids. Eur. Math. Soc. Tracts Math. 15, (2011)

  4. 4.

    Brown, R., Mucuk, O.: Covering groups of non-connected topological groups revisited. Math. Proc. Camb. Phill. Soc. 115, 97–110 (1994)

    Article  Google Scholar 

  5. 5.

    Brown, R., Spencer, C.B.: G-groupoids, crossed modules and the fundamental groupoid of a topological group. Proc. Konn. Ned. Akad. v. Wet. 79, 296–302 (1976)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Datuashvili, T.: Cohomology of internal categories in categories of groups with operations. In: Adamek, J., Lane, S.M. (eds.) Categoprical Topology and its Relation to Analysis, Algebra and Combinatorics, Proc. Conf. Categorical Topology, Prague 1988, pp. 270–283. World Scientific, Singapore (1989)

  7. 7.

    Datuashvili, T.: Cohomologically trivial internal categories in categories of groups with operations. Appl. Categor. Struct. 3(3), 221–237 (1995)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Datuashvili, T.: Whitehead homotopy equivalence and internal category equivalence of crossed modules in categories of groups with operations, Collected papers K-theory and Categorical Algebra. Proc. A. Razmadze Math Inst. Acad. Sci. Georgia 113, 3–30 (1995)

  9. 9.

    Datuashvili, T.: Kan extensions of internal functors. Nonconnected case. J. Pure Appl. Algebra 167, 195–202 (2002)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Hardie, K.A., Kamps, K.H., Kieboom, R.W.: A homotopy 2-groupoid of a topological space. Appl. Categor. Struct. 8, 209–234 (2000)

    Article  Google Scholar 

  11. 11.

    Hardie, K.A., Kamps, K.H., Kieboom, R.W.: A homotopy bigroupoid of a topological space. Appl. Categor. Struct. 9, 311–327 (2001)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Joyal, A. Street, R.: Braided monoidal categories, Macquarie Mathematics Report No. 860081 (1986). Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102, 20–78 (1993)

  13. 13.

    Laplaza, M.L.: Coherence for categories with group structure: an alternative approach. J. Algebra 84, 305–323 (1983)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Mac Lane, S.: Duality for groups. Bull. Am. Math. Soc. 56, 485–516 (1950)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Mac Lane, S.: Categories for the Working Mathematician, Graduate Texts in Mathematics, vol. 5. Springer, New York (1971); Second edition (1998)

  16. 16.

    Porter, T.: Extensions, crossed modules and internal categories in categories of groups with operations. Proc. Edinb. Math. Soc. 30, 373–381 (1987)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Schommer-Pries, C.J.: Central extensions of smooth 2-groups and a finite-dimensional string 2-group. Geom. Topol. 15(2), 609–676 (2011)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Sinh, H.X.: Gr-catégories, Université Paris 7, Thése de doctorat (1975)

  19. 19.

    Sinh, H.X.: Gr-catégories strictes. Acta Math. Vietnam. 3, 47–59 (1978)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Ulbrich, K.-H.: Kohärenz in Kategorien mit Gruppenstruktur. J. Algebra 72, 279–295 (1981)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Vitale, E.M.: Categorical Groups: A Special Topic in Higher Dimensional Categorical Algebra, September 9 (2014)

  22. 22.

    Vitale, E.M.: A Picard-Brauer exact sequence of categorical groups. J. Pure Appl. Algebra 175(1–3), 383–408 (2002)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Whitehead, J.H.C.: Combinatorial homotopy II. Bull. Am. Math. Soc. t. 55, 453–496 (1949)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

We would like to thank the editor Timothy Porter and the referee for their valuable comments and suggestions, and the editors in whole for the excellent editorial process that has been set up. The first author is grateful to Ercyies University (Kayseri, Turkey) and to Osman Mucuk for invitations and to the Rustaveli National Science Foundation for financial support, grant GNSF/ST09 730 3 -105.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Tamar Datuashvili.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Tim Porter.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Datuashvili, T., Mucuk, O. & Şahan, T. Groups up to congruence relation and from categorical groups to c-crossed modules. J. Homotopy Relat. Struct. 15, 625–640 (2020). https://doi.org/10.1007/s40062-020-00270-4

Download citation

Keywords

  • Group up to congruence relation
  • c-crossed module
  • action
  • Categorical group

Mathematics Subject Classification

  • 20L99
  • 20L05
  • 18D35