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A topos for continuous logic

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Daniel Figueroa and Benno van den Berg

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.

http://www.tac.mta.ca/tac/volumes/38/28/38-28.pdf

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