A Relative Monotone-Light Factorization System for Internal Groupoids

Given an exact category C , it is well known that the connected component reflector π 0 : Gpd ( C ) → C from the category Gpd ( C ) of internal groupoids in C to the base category 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 ( E ′ , M ∗ ) in Gpd ( C ), where M ∗ is the class of coverings in the sense of the corresponding Galois theory. However, when restricting to the case where C is an exact Mal’tsev category, we show that the so-called comprehensive factorization of regular epimorphisms in Gpd ( C ) is the relative monotone-light factorization system (in the sense of Chikhladze) in the category Gpd ( C ) corresponding to the connected component reflector, where E ′ is the class of final functors and M ∗ the class of regular epimorphic discrete fibrations.