Abstract
In a regular category \(\mathbb {E}\), the direct image along a regular epimorphism f of a preorder is not a preorder in general. In Set, its best preorder approximation is then its cocartesian image above f. In a regular category, the existence of such a cocartesian image above f of a preorder S is actually equivalent to the existence of the supremum \(R[f]\vee S\) among the preorders. We investigate here some conditions ensuring the existence of these cocartesian images or equivalently of these suprema. They apply to two very dissimilar contexts: any topos \(\mathbb {E}\) with suprema of countable chains of subobjects or any n-permutable regular category.
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Communicated by Maria Manuel Clementino.
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Bourn, D. On the Cocartesian Image of Preorders and Equivalence Relations in Regular Categories. Appl Categor Struct 30, 1153–1176 (2022). https://doi.org/10.1007/s10485-022-09686-w
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DOI: https://doi.org/10.1007/s10485-022-09686-w
Keywords
- Internal reflexive relation
- Preorder and equivalence relation
- Supremum of pairs of internal preorders and of equivalence relations
- Cocartesian image
- Regular epimorphism and regular category
- Congruence modular variety and category
- Congruence n-permutable variety and category
- Elementary topos