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- W2768606178 abstract "Given an exact category $mathcal{C}$, it is well known that the connected component reflector $pi_0colonmathsf{Gpd}(mathcal{C})tomathcal{C}$ from the category $mathsf{Gpd}(mathcal{C})$ of internal groupoids in $mathcal{C}$ to the base category $mathcal{C}$ 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 $(mathcal{E}',mathcal{M}^*)$ in $mathsf{Gpd}(mathcal{C})$, where $mathcal{M}^*$ is the class of coverings in the sense of the corresponding Galois theory. However, when restricting to the case where $mathcal{C}$ is an exact Mal'tsev category, we show that the so-called comprehensive factorization of regular epimorphisms in $mathsf{Gpd}(mathcal{C})$ is the relative monotone-light factorisation system (in the sense of Chikhladze) in the category $mathsf{Gpd}(mathcal{C})$ corresponding to the connected component reflector, where $mathcal{E}'$ is the class of final functors and $mathcal{M}^*$ the class of regular epimorphic discrete fibrations." @default.
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- W2768606178 date "2017-11-28" @default.
- W2768606178 modified "2023-09-27" @default.
- W2768606178 title "A relative monotone-light factorisation system for internal groupoids" @default.
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