- Hacettepe Journal of Mathematics and Statistics
- Vol: 49 Issue: 3
- Covering morphisms of topological internal groupoids
Covering morphisms of topological internal groupoids
Authors : Osman Mucuk, Hürmet Fulya Akiz
Pages : 1020-1029
Doi:10.15672/hujms.467559
View : 8 | Download : 3
Publication Date : 2020-06-02
Article Type : Research
Abstract :Let $X$ be a topological group with operations whose underlying space has a universal cover. Then the fundamental groupoid $\pi X$ becomes a topological internal groupoid, i.e., an internal groupoid in the category of topological groups. In this paper, we prove that the slice category $\text{Cov}_{sTC}/X$ of covering morphisms $p:\tilde{X}\rightarrow X$ of topological groups with operations in which $\tilde{X}$ has also a universal cover and the category $\text{Cov}_{Gpd(TC)}/\pi X$ of covering morphisms $q:\tilde{G}\rightarrow \pi X $ of topological internal groupoids based on $\pi X$ are equivalent. We also prove that for a topological internal groupoid $G$, the category $\text{Cov}_{Gpd(TC)}/G$ of covering morphisms of topological internal groupoids based on $G$ and the category $\text{ACT}_{Gpd(TC)}/G$ of topological internal groupoid actions of $G$ on topological groups with operations are equivalent.Keywords : Covering groupoid, internal groupoids, topological internal groupoids