Covering theory for strict 2-groupoids via actions, quotients, and crossed modules over groupoids

Sedat Temel; Sedat Temel

doi:10.3934/math.20261095

AIMS Mathematics

2026, Volume 11, Issue 8: 27370-27395. doi: 10.3934/math.20261095

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Research article

Covering theory for strict 2-groupoids via actions, quotients, and crossed modules over groupoids

Sedat Temel ^,

Department of Mathematics, Recep Tayyip Erdogan University, Rize 53100, Türkiye

Received: 19 June 2026 Revised: 11 August 2026 Accepted: 13 August 2026 Published: 31 August 2026
MSC : 20L05, 18D35, 18B40, 18N10

This paper establishes a direct correspondence between the geometric covering theory of strict 2-groupoids and the algebraic lifting theory of crossed modules over groupoids. Working entirely within an object-preserving framework, we prove a chain of categorical equivalences linking strict 2-groupoid actions, covering strict 2-groupoids, liftings of crossed modules, and covering crossed modules. Additionally, we introduce a quotient theory for strict 2-groupoids via normal 2-group bundles. We demonstrate that this quotient construction preserves the covering property and naturally induces canonical lifting crossed modules, providing a unified perspective on these geometric and algebraic structures.
- strict 2-groupoid,
- crossed module,
- covering,
- lifting,
- quotient groupoid
Citation: Sedat Temel. Covering theory for strict 2-groupoids via actions, quotients, and crossed modules over groupoids[J]. AIMS Mathematics, 2026, 11(8): 27370-27395. doi: 10.3934/math.20261095

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Abstract

This paper establishes a direct correspondence between the geometric covering theory of strict 2-groupoids and the algebraic lifting theory of crossed modules over groupoids. Working entirely within an object-preserving framework, we prove a chain of categorical equivalences linking strict 2-groupoid actions, covering strict 2-groupoids, liftings of crossed modules, and covering crossed modules. Additionally, we introduce a quotient theory for strict 2-groupoids via normal 2-group bundles. We demonstrate that this quotient construction preserves the covering property and naturally induces canonical lifting crossed modules, providing a unified perspective on these geometric and algebraic structures.

References

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