Abstract
The aim of this paper is to characterize the notion of internal category (groupoid) in the category of Leibniz algebras and investigate the properties of well-known notions such as covering groupoid and groupoid operations (actions) in this category. Further, for a fixed internal groupoid $G$, we prove that the category of covering groupoids of $G$ and the category of internal groupoid actions of $G$ on Leibniz algebras are equivalent. Finally we interpret the corresponding notion of covering groupoids in the category of crossed modules of Leibniz algebras.
Highlights
Covering groupoids have an important role in the applications of groupoids
The aim of this paper is to characterize the notion of internal category in the category of Leibniz algebras and investigate some properties of well-known notions such as covering groupoids and groupoid operations in this category
For a ...xed internal groupoid G in the category of Leibniz algebras, we prove that the category of covering groupoids of G and the category of internal groupoid actions of G on Leibniz algebras are equivalent
Summary
Covering groupoids have an important role in the applications of groupoids (see for example [3] and [14]). In [26] authors have interpreted in the category of crossed modules the notion of action of a group-groupoid on a group via a group homomorphism and introduced the notion of lifting of a crossed module over groups Further they showed some results on liftings of crossed modules and proved that the category of liftings of crossed modules, the category of covering crossed modules and the category of group-groupoid actions are equivalent. In order to interpret the notion of liftings in the category of crossed modules over Leibniz algebras one needs the detailed de...nitions and properties of internal action groupoid, covering groupoid, covering crossed module in this category. Using the equivalence between the categories of internal groupoids in the category of Leibniz algebras and crossed modules in the category of Leibniz algebras, we interpret the notion of covering in the category of crossed modules in the category of Leibniz algebras
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