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


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.

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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.

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Correspondence to Tamar Datuashvili.

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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).

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

Mathematics Subject Classification

  • 20L99
  • 20L05
  • 18D35