A Relative Monotone-Light Factorization System for Internal Groupoids

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