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.
Keywords: Continuous logic, metric spaces, categorical logic, hyperdoctrines, Grothendieck toposes
2020 MSC: 03C66,03G30,18F10
Theory and Applications of Categories, Vol. 38, 2022, No. 28, pp 1108-1135.
Published 2022-09-07.
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