A Relative Monotone-Light Factorization System for Internal Groupoids

Given an exact category $${\mathcal {C}}$$C, it is well known that the connected component reflector $$ \pi _0 :\mathsf {Gpd}(\mathcal {C}) \rightarrow \mathcal {C}$$π0:Gpd(C)→C from the category $$\mathsf {Gpd}(\mathcal {C})$$Gpd(C) of internal groupoids in $$\mathcal {C}$$C to the base category $$\mathcal {C}$$C is semi-left-exact. In this article we investigate the existence of a monotone-light factorization system associated with this reflector. We show that, in general, there is no monotone-light factorization system $$(\mathcal {E}',\mathcal {M}^*)$$(E′,M∗) in $$\mathsf {Gpd}$$Gpd($$\mathcal {C}$$C), where $$\mathcal {M}^*$$M∗ is the class of coverings in the sense of the corresponding Galois theory. However, when restricting to the case where $$\mathcal {C}$$C is an exact Mal’tsev category, we show that the so-called comprehensive factorization of regular epimorphisms in $$\mathsf {Gpd}$$Gpd($$\mathcal {C}$$C) is the relative monotone-light factorization system (in the sense of Chikhladze) in the category $$\mathsf {Gpd}$$Gpd($$\mathcal {C}$$C) corresponding to the connected component reflector, where $$\mathcal {E}'$$E′ is the class of final functors and $$ \mathcal {M}^*$$M∗ the class of regular epimorphic discrete fibrations.