In this present work, we present the concept of a crossed module over generalized groups and we call it a “generalized crossed module”. We also define a generalized group-groupoid. Furthermore we show that the category of generalized crossed modules is equivalent to that of generalized group-groupoids whose object sets are abelian generalized group. | Turk J Math (2017) 41: 1535 – 1551 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Generalized crossed modules and group-groupoids 1 1,∗ ¨ ˙ ˙ ¸ EN1 Mustafa Habil GURSOY , Hatice ASLAN 2 , Ilhan IC ˙ Department of Mathematics, Faculty of Science and Arts, In¨ on¨ u University, Malatya, Turkey 2 Department of Mathematics, Faculty of Science,Fı rat University, Elazı˘ g, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: In this present work, we present the concept of a crossed module over generalized groups and we call it a “generalized crossed module”. We also define a generalized group-groupoid. Furthermore, we show that the category of generalized crossed modules is equivalent to that of generalized group-groupoids whose object sets are abelian generalized group. Key words: Groupoid, crossed module, generalized group 1. Introduction The generalized group, first defined by Molaei [13] in 1999, is an interesting generalization of groups. While there is only one identity element in a group, each element in a generalized group has a unique identity element. With this property, every group is a generalized group. After Molaei gave the definition of a generalized group, this concept was studied in terms of algebraic, topological, and differentiable in large various areas of mathematics [1, 2, 8, 12–15]. Another algebraic concept covered in the present study is the crossed module. The concept of crossed module was defined over groups by Whitehead [19]. Afterwards, crossed modules were studied extensively in many areas of mathematics by defining them also over other algebraic structures [3, 6, 16, 17]. We also define the concept of crossed module over generalized groups (called the generalized crossed module). A generalized crossed module is a generalization of the crossed module over groups. We construct the category of .
