Given an exact category C, it is well known that the connected component reflector π0: 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.

A Relative Monotone-Light Factorization System for Internal Groupoids

Cigoli A. S.
;
2018-01-01

Abstract

Given an exact category C, it is well known that the connected component reflector π0: 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.
2018
26
5
931
942
Factorization system; Internal groupoids; Mal’tsev category; Monotone-light factorization
Cigoli A.S.; Everaert T.; Gran M.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/1837411
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