Crossed modules, double group-groupoids and crossed squares
Sedat Temel, Tun\c{c}ar \c{S}ahan, Osman Mucuk
TL;DR
This paper establishes a categorical equivalence between crossed modules over group-groupoids (2-groups) and double group-groupoids, expanding the understanding and examples of these algebraic structures.
Contribution
It introduces the notion of crossed modules over group-groupoids via split extensions and proves their equivalence to double group-groupoids within the category of group-groupoids.
Findings
Categorical equivalence between crossed modules and double group-groupoids
Construction of new examples of double group-groupoids
Extension of the theory of 2-groups and double groupoids
Abstract
In this paper using split extensions of group-groupoids we obtain the notion of crossed modules over group-grouoids which are also called 2-groups and we prove a categorical equivalence of these types of crossed modules and double group-groupoids which are internal to the category of group-groupoids. This equivalence enables us to produce more examples of double groupoids.
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