We consider the categorical equivalence between crossed modules over groupoids and double groupoids with thin structures, and by this equivalence, we prove how normality and quotient concepts are related in these two categories and give some examples of these objects. | Turk J Math (2018) 42: 2336 – 2347 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Normality and quotient in crossed modules over groupoids and double groupoids Osman MUCUK1,∗,, Serap DEMİR2 , Department of Mathematics, Faculty of Science, Erciyes University, Kayseri, Turkey 2 Department of Mathematics, Faculty of Science, Erciyes University, Kayseri, Turkey 1 Received: • Accepted/Published Online: • Final Version: Abstract: We consider the categorical equivalence between crossed modules over groupoids and double groupoids with thin structures, and by this equivalence, we prove how normality and quotient concepts are related in these two categories and give some examples of these objects. Key words: Quotient crossed module, double groupoid, quotient double groupoid 1. Introduction The concept of a crossed module over groups introduced by Whitehead in [23, 24] in the investigation of the properties of second relative homotopy groups for topological spaces, which can be viewed as a 2-dimensional group [3], has been widely used in homotopy theory [5], the theory of identities among relations for group presentations [6], algebraic K-theory [16], and homological algebra [15, 17]. See [5, p. 49], for some discussion of the relation of crossed modules to crossed squares and so to homotopy 3-types. The categorical equivalence of crossed modules over groups and group-groupoids, which are internal groupoids in the category of groups and widely used in the literature under the names 2-groups[2], G -groupoids, or group objects in the category of groupoids [8], was proved by Brown and Spencer in [8, Theorem 1], and then some important results were obtained by means of this equivalence. For example, recently normal and quotient objects in these two categories were compared and the corresponding objects in the category of group-groupoids were characterized in [18]. .
