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A topos for continuous logic
Theory and applications of categories, Tome 38 (2022), pp. 1108-1135
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We suggest an ordering for the predicates in continuous logic so that the semantics of continuous logic can be formulated as a hyperdoctrine. We show that this hyperdoctrine can be embedded into the hyperdoctrine of subobjects of a suitable Grothendieck topos. For this embedding we use a simplification of the hyperdoctrine for continuous logic, whose category of equivalence relations is equivalent to the category of complete metric spaces and uniformly continuous maps.

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Classification : 03C66, 03G30, 18F10
Keywords: Continuous logic, metric spaces, categorical logic, hyperdoctrines, Grothendieck toposes 
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Daniel Figueroa; Benno van den Berg. A topos for continuous logic. Theory and applications of categories, Tome 38 (2022), pp. 1108-1135. http://geodesic.mathdoc.fr/item/TAC_2022_38_a27/