Double groupoids and 2-groupoids in regular Mal'tsev categories

Gran, Marino;Egner, Nadja
(2025) Applied Categorical Structures — Vol. 33 (2025)

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We prove that the category 2-Grpd (C) of internal 2-groupoids is a Birkhoff subcategory of the category Grpd^2(C) of double groupoids in a regular Mal'tsev category C with finite colimits. In particular, when C is a Mal'tsev variety of universal algebras, the category 2-Grpd(C) is also a Mal'tsev variety, of which we describe the corresponding algebraic theory. When C is a naturally Mal'tsev category, the reflector from Grpd^2(C) to 2-Grpd(C) has an additional property related to the commutator of equivalence relations. We prove that the category 2-Grpd(C) is semi-abelian when C is semi-abelian, and then provide sufficient conditions for 2-Grpd(C) to be action representable.
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Gran, M., & Egner, N. (2025). Double groupoids and 2-groupoids in regular Mal’tsev categories. Applied Categorical Structures, 33. https://hdl.handle.net/2078.5/267830 (Original work published 2025)