This enables one to prove that thecategory of the covering spaces of an H-group (X, x0) is equivalent tothe category of covering groupoids of the weak categorical group π1(X)
📌 3 Key Takeaways
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Objective & Hypothesis:If X is a topological group, then its fundamental groupoid π1(X) is agroup-groupoid which is a group object in the category of groupoids.Further if X is a path connected topological group which has a simplyconnected cover, then the category of covering groups of X and thecategory of covering groupoids of π1(X) are equivalent.
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Key Findings & Data:In this paperwe prove that if (X, x0) is an H-group, then the fundamental groupoidπ1(X) is a weak categorical group.
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Conclusion & Impact:This enables one to prove that thecategory of the covering spaces of an H-group (X, x0) is equivalent tothe category of covering groupoids of the weak categorical group π1(X)
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:H-group · covering groupoid and categorical group
Abstract
If X is a topological group, then its fundamental groupoid π1(X) is agroup-groupoid which is a group object in the category of groupoids.Further if X is a path connected topological group which has a simplyconnected cover, then the category of covering groups of X and thecategory of covering groupoids of π1(X) are equivalent. In this paperwe prove that if (X, x0) is an H-group, then the fundamental groupoidπ1(X) is a weak categorical group. This enables one to prove that thecategory of the covering spaces of an H-group (X, x0) is equivalent tothe category of covering groupoids of the weak categorical group π1(X)
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