Maps. I: Relative to a factorisation system
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.
scientific article; zbMATH DE number 762336
scientific article