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

We prove that the category 2-$\mathrm{Grpd}(\mathscr{C})$ of internal $2$-groupoids is a Birkhoff subcategory of the category $\mathrm{Grpd}^2(\mathscr{C})$ of double groupoids in a regular Mal'tsev category $\mathscr{C}$ with finite colimits. In particular, when $\mathscr{C}$ is a Mal'tsev variety of universal algebras, the category 2-$\mathrm{Grpd}(\mathscr{C})$ 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 $\mathrm{Grpd}^2(\mathscr{C})$ to 2-$\mathrm{Grpd}(\mathscr{C})$ has an additional property related to the commutator of equivalence relations. We prove that the category 2-$\mathrm{Grpd}(\mathscr{C})$ is semi-abelian when $\mathscr{C}$ is semi-abelian, and then provide sufficient conditions for 2-$\mathrm{Grpd}(\mathscr{C})$ to be action representable.