$N$-fold groupoids and $n$-groupoids in regular Mal’tsev categories

Mal’tsev categories are a central concept in categorical algebra. In [2], a Mal’tsev category is defined as a finitely complete category $ℂ$ in which any internal reflexive relation is an internal equivalence relation. A regular category $ℂ$ is a Mal’tsev category if and only if the composition of internal equivalence relations on any object in $ℂ$ is commutative. Examples of Mal’tsev categories are given by any abelian category, the dual of any elementary topos, and any semi-abelian category such as the categories of groups and of Heyting semi-lattices.

Internal structures in a Mal’tsev category are well-behaved. Whereas it is in general not true that the category $\mathrm{Cat}(ℂ)$ of internal categories in a (Barr-)exact category $ℂ$ is again exact, it is shown in [4] that $\mathrm{Cat}(ℂ)$ is an exact Mal’tsev category whenever $ℂ$ is exact Mal’tsev. Moreover, any internal reflexive graph in a Mal’tsev category admits at most one internal category structure, which yields automatically an internal groupoid.

The category ${\mathrm{Cat}}^n(ℂ)$ of internal $n$-fold categories in a finitely complete category $ℂ$ can be defined recursively as the category $\mathrm{Cat}({\mathrm{Cat}}^{n-1}(ℂ))$ of internal categories in the category ${\mathrm{Cat}}^{n-1}(ℂ)$ of internal $(n-1)$-fold categories in $ℂ$. In [1], also the category $n\text{-}\mathrm{Cat}(ℂ)$ of internal $n$-categories in $ℂ$ is defined recursively. For $ℂ=\mathrm{Set}$ being the category of sets, this yields the usual definitions of $n$-fold category and $n$-category, respectively.

These recursive descriptions help us to generalize our results in [3] on the relation between ${\mathrm{Cat}}^2(ℂ)={\mathrm{Grpd}}^2(ℂ)$ and $2\text{-}\mathrm{Cat}(ℂ)=2\text{-}\mathrm{Grpd}(ℂ)$, where $ℂ$ is a regular Mal’tsev category with finite colimits. We show that $n\text{-}\mathrm{Grpd}(ℂ)$ is a Birkhoff subcategory, i.e. a reflective subcategory closed under subobjects and quotients, of ${\mathrm{Grpd}}^n(ℂ)$. In consequence, $n\text{-}\mathrm{Grpd}(ℂ)$ is regular Mal’tsev, exact Mal’tsev or semi-abelian whenever $ℂ$ is so. Using the results in [5], we can also show that $n\text{-}\mathrm{Grpd}(ℂ)$ is action representable whenever $ℂ$ is semi-abelian, action representable, algebraically coherent and with normalizers.