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- W2963655342 abstract "Given an exact category $${mathcal {C}}$$ , it is well known that the connected component reflector $$ pi _0 :mathsf {Gpd}(mathcal {C}) rightarrow mathcal {C}$$ 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 factorization system associated with this reflector. We show that, in general, there is no monotone-light factorization 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 factorization 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.
- W2963655342 created "2019-07-30" @default.
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- W2963655342 date "2018-02-12" @default.
- W2963655342 modified "2023-09-23" @default.
- W2963655342 title "A Relative Monotone-Light Factorization System for Internal Groupoids" @default.
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- W2963655342 doi "https://doi.org/10.1007/s10485-018-9515-5" @default.
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