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

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