Double Groupoids and 2-Groupoids in Regular Mal’tsev Categories

We prove that the category 2-\(\textrm{Grpd}(\mathscr {C})\) of internal 2-groupoids is a Birkhoff subcategory of the category \(\textrm{Grpd}^2(\mathscr {C})\) of double groupoids in a regular Mal’tsev category \(\mathscr {C}\) with finite colimits, and we provide a simple description of the reflector. In particular, when \(\mathscr {C}\) is a Mal’tsev variety of universal algebras, the category 2-\(\textrm{Grpd}(\mathscr {C})\) is also a Mal’tsev variety, of which we describe the corresponding algebraic theory. When \(\mathscr {C}\) is a naturally Mal’tsev category, the reflector from \(\textrm{Grpd}^2(\mathscr {C})\) to 2-\(\textrm{Grpd}(\mathscr {C})\) has an additional property related to the commutator of equivalence relations. We prove that the category 2-\(\textrm{Grpd}(\mathscr {C})\) is semi-abelian when \(\mathscr {C}\) is semi-abelian, and then provide sufficient conditions for 2-\(\textrm{Grpd}(\mathscr {C})\) to be action representable.