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Dergiler / Hacettepe Journal of Mathematics and Statistics / 2020 / Cilt: 49 - Sayı: 3

Covering morphisms of topological internal groupoids

Osman MUCUK
H. Fulya AKIZ
Sayfa
1020–1029
Yayın Tarihi
2022-10-18
DOI
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⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

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.

📌 3 Maddede Bu Makale

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Amaç & Hipotez:Let $X$ be a topological group with operations whose underlying space has a universal cover.
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Temel Bulgular & Veri:Then the fundamental groupoid $\pi X$ becomes a topological internal groupoid, i.e., an internal groupoid in the category of topological groups.
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Sonuç & Katkı: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.
🔬Tür:Araştırma Makalesi
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Covering groupoid · internal groupoids · topological internal groupoids

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.

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Bu Makaleye Atıf Yap

MUCUK, O., AKIZ, H. F. (2022). Covering morphisms of topological internal groupoids. Hacettepe Journal of Mathematics and Statistics, Cilt: 49 - Sayı: 3, 1020–1029. https://ajindex.com/articles/529652

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