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Constructive sheaf models of type theory

Published online by Cambridge University Press:  18 November 2021

Thierry Coquand*
Computer Science Department, Chalmers University and University of Gothenburg, Gothenburg, Sweden
Fabian Ruch
Computer Science Department, Chalmers University and University of Gothenburg, Gothenburg, Sweden
Christian Sattler
Computer Science Department, Chalmers University and University of Gothenburg, Gothenburg, Sweden
*Corresponding author. Email:
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We provide a constructive version of the notion of sheaf models of univalent type theory. We start by relativizing existing constructive models of univalent type theory to presheaves over a base category. Any Grothendieck topology of the base category then gives rise to a family of left-exact modalities, and we recover a model of type theory by localizing the presheaf model with respect to this family of left-exact modalities. We provide then some examples.

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