Groups up to congruence relation and from categorical groups to c-crossed modules
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Groups up to congruence relation and from categorical groups to c-crossed modules 3 Tamar Datuashvili1 · Osman Mucuk2 · Tunçar Sahan ¸
Received: 5 May 2020 / Accepted: 25 October 2020 / Published online: 21 November 2020 © Tbilisi Centre for Mathematical Sciences 2020
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. Keywords Group up to congruence relation · c-crossed module · action · Categorical group Mathematics Subject Classification 20L99 · 20L05 · 18D35
1 Introduction Our aim was to obtain for categorical groups an analogous description in terms of certain crossed module type objects as we have for G-groupoids obtained by Brown
Communicated by Tim Porter.
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Tamar Datuashvili [email protected] Osman Mucuk [email protected] Tunçar Sahan ¸ [email protected]
1
A. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University, 6 Tamarashvii Str., 0177 Tbilisi, Georgia
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Department of Mathematics, Erciyes University, 38039 Kayseri, Turkey
3
Department of Mathematics, Aksaray University, 68100 Aksaray, Turkey
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and Spencer [5], which are strict categorical groups, or equivalently, group-groupoids or internal categories in the category of groups. By a categorical group we mean a coherent 2-group in the sense of Baez and Lauda [1]. It is important to note that it is well known that a categorical group is equivalent to a strict categorical group [1,12,19], but we do not have an equivalence between the corresponding categories. This idea brought us to a new notion of group up to congruence relation. In this way we came to the definition of c-group and the corresponding category. Then we defined actions in this category and introduced the notion of c-crossed module. Among this kind of objects we distinguished connected, strict and special c-crossed modules denoted as cssc-crossed modules. We proved that every categorical group gives rise to a cssccrossed module. The prototypes of all the new concepts introduced in this paper are those obtained from categorical groups. In the sequel to this paper we will prove that there is an equivalence between the category of categorical groups and the category of cssc-crossed modules. We hope that this result will give a chance to consider for categorical groups the problems analogous to those considered and solved in the case of strict categorical groups in terms of group-groupoids and internal categories in [4,6–9]. We would like to thank one of the editors of the journal who
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