A Relative Monotone-Light Factorization System for Internal Groupoids

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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.

Keywords

Factorization system Monotone-light factorization Internal groupoids Mal’tsev category 

Mathematics Subject Classification

18A40 18A32 18A22 18D35 08C05 18E10 

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Authors and Affiliations

  1. 1.INdAM Cofund Marie Curie fellow. Institut de recherche en mathématique et physiqueUniversité catholique de LouvainLouvain–la–NeuveBelgium
  2. 2.Dipartimento di MatematicaUniversità degli Studi di MilanoMilanoItaly
  3. 3.Institut de recherche en mathématique et physiqueUniversité catholique de LouvainLouvain–la–NeuveBelgium

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