Maps. I: Relative to a factorisation system
type: Item
instance of: scholarly article
title: Maps. I: Relative to a factorisation system
zbMATH Open document ID: 0829.18002
DOI: 10.1016/0022-4049(94)00054-M
author: Dusko Pavlovic
published in: Journal of Pure and Applied Algebra
publication date: 1995-07-03T00:00:00Z
review text: The author considers a finitely complete category \(\mathcal C\) with a stable \(({\mathcal E}, {\mathcal M})\)-factorization system for morphisms (without any epi-mono conditions), and defines the bicategory \({\mathcal R} = \text{Rel} (C)\) of \(\mathcal M\)-relations, with 1-cells \(r : A \to B\) given by arrows \(r : R \to A \times B\) in \(\mathcal M\). According to \textit{F. W. Lawvere}'s definition [cf. Rend. Semin. Mat. Fis. Milano 43 (1973), 135-166 (1974; Zbl 0335.18006)], \(r\) is a map if it has a right adjoint. It is not at all obvious in the present context that this right adjoint must be given by the converse relation of \(r\). With this observation one is able to define the ordinary category \(\text{Map} ({\mathcal R})\), whose arrows are isomorphic classes of maps in \(\mathcal R\). The author shows that the graph functor \(G : {\mathcal C} \to \text{Map} (\text{Rel} ({\mathcal C}))\) is faithful iff \(\mathcal E\) is a class of epimorphisms, and it is full and faithful (in fact: an isomorphism of categories) iff \(\mathcal E\) is a class of regular epimorphisms. Hence, on the one hand he gives a-posteriori justification for the restriction to regular categories as in earlier papers [starting with \textit{J. Meisen}, Commun. Algebra 1, 377-401 (1974; Zbl 0286.18002)]; on the other hand, he provides plenty of evidence that such a restriction does not suit potential applications in logic.
reviewed by: Walter Tholen
Mathematics Subject Classification ID: 18B10
Mathematics Subject Classification ID: 18A32
Mathematics Subject Classification ID: 18D05
zbMATH DE Number: 762336
zbMATH Keywords: relations
zbMATH Keywords: factorization system
zbMATH Keywords: bicategory
zbMATH Keywords: maps
MaRDI profile type: Publication
cites work: Exact categories and categories of sheaves
cites work: Q3682648
cites work: Introduction to bicategories
cites work: Variation through enrichment
cites work: Q3795831
cites work: Bicategories of spans and relations
cites work: Q3986541
cites work: Cartesian bicategories. I
cites work: Q3961638
cites work: Erratum
cites work: Q3995720
cites work: Q3491793
cites work: Q3978980
cites work: Relations in categories
cites work: Factorization systems as Eilenberg-Moore algebras
cites work: Metric spaces, generalized logic, and closed categories
cites work: Constructive natural deduction and its ‘ω-set’ interpretation
cites work: On Bicategories of Relations and Pullback Spans
cites work: Q4282578
cites work: On completeness and cocompleteness in and around small categories
cites work: Q3342724
cites work: Cauchy characterization of enriched categories
cites work: Conspectus of variable categories
cites work: Constructivism in mathematics. An introduction. Volume I
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description: scientific article; zbMATH DE number 762336