We introduce distributive maps between lattices and consider the categorical assumption that distributive idempotents split. We explore this assumption in the context of a categorical axiomatization of the category of locales. The assumption is shown to be stable under groupoids (this includes slice stability) and we further show that it implies that triquotient surjections are effective descent morphisms. This result follows even without assuming that the underlying (axiomatized) category of locales has coequalizers.
Keywords: Locale, topos, categorical logic, powerlocales, order enriched, distributive lattice, axioms
2020 MSC: 06D22, 03G30
Theory and Applications of Categories, Vol. 40, 2024, No. 9, pp 278-300.
Published 2024-04-15.
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