A relative monotone-light factorisation system for internal groupoids

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Given an exact category C, it is well known that the connected component reflector π₀(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 factorisation system associated with this reflector. We show that, in general, there is no monotone-light factorisation 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 factorisation 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.