A new Galois structure in the category of internal preorders. (English) Zbl 1435.18009

Summary: Let \(\mathsf{PreOrd}(\mathbb{C})\) be the category of internal preorders in an exact category \(\mathbb{C}\). We show that the pair (\(\mathsf{Eq}(\mathbb{C}),\,\mathsf{ParOrd}(\mathbb{C})\)) is a pretorsion theory in \(\mathsf{PreOrd}(\mathbb{C})\), where \(\mathsf{Eq}(\mathbb{C})\) and \(\mathsf{ParOrd}(\mathbb{C})\) are the full subcategories of internal equivalence relations and of internal partial orders in \(\mathbb{C}\), respectively. We observe that \(\mathsf{ParOrd}(\mathbb{C})\) is a reflective subcategory of \(\mathsf{PreOrd}(\mathbb{C})\) such that each component of the unit of the adjunction is a pullback-stable regular epimorphism. The reflector \(F\): \(\mathsf{PreOrd}(\mathbb{C}) \to \mathsf{ParOrd}(\mathbb{C})\) turns out to have stable units in the sense of Cassidy, Hébert and Kelly, thus inducing an admissible categorical Galois structure. In particular, when \(\mathbb{C}\) is the category \(\mathsf{Set}\) of sets, we show that this reflection induces a monotone-light factorization system (in the sense of [A. Carboni et al., Appl. Categ. Struct. 5, No. 1, 1–58 (1997; Zbl 0866.18003)]) in \(\mathsf{PreOrd(Set)}\). A topological interpretation of our results in the category of Alexandroff-discrete spaces is also given, via the well-known isomorphism between this latter category and \(\mathsf{PreOrd(Set)}\).


18E50 Categorical Galois theory
18A32 Factorization systems, substructures, quotient structures, congruences, amalgams
18B35 Preorders, orders, domains and lattices (viewed as categories)
18E40 Torsion theories, radicals
06A15 Galois correspondences, closure operators (in relation to ordered sets)


Zbl 0866.18003
Full Text: arXiv Link


[1] M. Barr, Exact Categories, in: Exact Categories and Categories of Sheaves, in: Lect. Notes Math.236, Springer (1971) pp. 1-120. · Zbl 0223.18009
[2] D. Bourn, The shift functor and the comprehensive factorization for internal groupoids,Cahiers Topologie G´eom. Diff´erentielle Cat´eg.28197-226 (1987).
[3] D. Bourn and M. Gran, Torsion theories in homological categories,J. Algebra305 (1) (2006) 18-47. · Zbl 1123.18009
[4] A. Carboni, G. Janelidze, G.M. Kelly and R. Par´e, On localization and stabilization for factorization systems,Appl. Categ. Struct.5(1997) 1-58. · Zbl 0866.18003
[5] A. Carboni, G.M. Kelly and M.C. Pedicchio, Some remarks on Maltsev and Goursat categories,Appl. Categ. Structures,1(1993) 385-421. · Zbl 0799.18002
[6] C. Cassidy, M. H´ebert and G.M. Kelly, Reflective subcategories, localizations and factorizations systems,J. Aust. Math. Soc.38(1985) 287-329. · Zbl 0573.18002
[7] S.C. Dickson, A torsion theory for abelian categories,Trans. Amer. Math. Soc.21 (1966) 223-235. · Zbl 0138.01801
[8] C. Ehresmann, Sur une notion g´en´erale de cohomologie,C. R. Acad. Sci. Paris259 (1964) 2050-2053. · Zbl 0242.18015
[9] A. Facchini and C. Finocchiaro, Pretorsion theories, stable category and preordered sets, Ann. Mat. Pura Appl. (4) (2019) https://doi.org/10.1007/s10231-019-00912-2.
[10] A. Facchini, C. Finocchiaro and M. Gran, Pretorsion theories in general categories, preprint arXiv:1908.03546 (2019).
[11] M. Gran, D. Rodelo, and I. Tchoffo Nguefeu, Some remarks on connectors and groupoids in Goursat categories,Logical Methods in Computer Science,13(3:14) (2017) 1-12. · Zbl 1453.18004
[12] M. Grandis and G. Janelidze, From torsion theories to closure operators and factorization systems,Categ. Gen. Algebr. Struct. Appl., 2020,12(1) 89-121.
[13] G. Janelidze, Pure Galois theory in categories,J. Algebra,132(1990) 270-286. · Zbl 0702.18006
[14] G. Janelidze and G.M. Kelly, Galois theory and a general notion of central extension J. Pure Appl. Algebra2(1994) 135-161. · Zbl 0813.18001
[15] G. Janelidze, L. M´arki and W. Tholen, Locally semi-simple coverings,J. Pure Appl. Algebra,128(1998) 281-289. · Zbl 0927.18004
[16] G. Janelidze and M. Sobral, Finite preorders and topological descent I,J. Pure Appl. Algebra175(2002) 187-205. · Zbl 1018.18004
[17] N. Martins-Ferreira, D. Rodelo and T. Van der Linden, An observation onnpermutability,Bull. Belgian Math. Soc. Simon Stevin, 21 (2014) 223-230. · Zbl 1301.08014
[18] S. Mantovani, Torsion theories for crossed modules, invited talk at the “Workshop on category theory and topology”, September 2015, Universit´e catholique de Louvain.
[19] S. Mantovani, Semilocalizations of exact and lextensive categories,Cah. Topol. G´eom. Diff´erent. Cat´eg.39(1998) 27-44.
[20] J. Xarez, Internal monotone-light factorization for categories via preorders,Theory Appl. Categories13(2004) 235-251. · Zbl 1059.18002
[21] J.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.