# Actions of internal groupoids in the category of Leibniz algebras

## TL;DR

This paper characterizes internal groupoids within Leibniz algebras, explores their properties, and establishes an equivalence between covering groupoids and actions, extending the theory to crossed modules.

## Contribution

It introduces a framework for internal groupoids in Leibniz algebras and proves an equivalence between covering groupoids and actions, extending existing algebraic structures.

## Key findings

- Equivalence between covering groupoids and internal groupoid actions
- Characterization of internal categories and groupoids in Leibniz algebras
- Extension of covering groupoid concepts to crossed modules

## 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.