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Constructive sheaf models of type theory

Published online by Cambridge University Press:  18 November 2021

Thierry Coquand Open the ORCID record for Thierry Coquand [Opens in a new window] ,
Fabian Ruch  and
Christian Sattler Open the ORCID record for Christian Sattler [Opens in a new window]
Thierry Coquand*
Affiliation:
Computer Science Department, Chalmers University and University of Gothenburg, Gothenburg, Sweden
Fabian Ruch
Affiliation:
Computer Science Department, Chalmers University and University of Gothenburg, Gothenburg, Sweden
Christian Sattler
Affiliation:
Computer Science Department, Chalmers University and University of Gothenburg, Gothenburg, Sweden
*
*Corresponding author. Email: Thierry.Coquand@cse.gu.se
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Abstract

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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.

Keywords

Dependent type theoryhomotopy type theorysheaf modelsconstructive models of univalenceleft-exact modalities
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 31 , Special Issue 9: In Homage to Martin Hofmann , October 2021 , pp. 979 - 1002
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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