Lawvere-Tierney sheafification in Homotopy Type Theory

Kevin Quirin 1 Nicolas Tabareau 1
1 ASCOLA - Aspect and composition languages
Inria Rennes – Bretagne Atlantique , Département informatique - EMN, LINA - Laboratoire d'Informatique de Nantes Atlantique
Abstract : Sheafification is a popular tool in topos theory which allows to extend the internal logic of a topos with new principles. One of its most famous applications is the possibility to transform a topos into a boolean topos using the dense topology, which corresponds in essence to Gödel's double negation translation. The same construction has not been developed in Martin-Löf type theory because of a mismatch between topos theory and type theory. This mismatch has been fixed recently by considering homotopy type theory, an extension of Martin-Löf type theory with new principles inspired by category theory and homotopy theory, and which corresponds closely to higher toposes. In this paper, we give a computer-checked construction of Lawvere-Tierney sheafification in homotopy type theory.
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Kevin Quirin, Nicolas Tabareau. Lawvere-Tierney sheafification in Homotopy Type Theory. Journal of Formalized Reasoning, ASDD-AlmaDL, 2016, 9 (2), ⟨10.6092/issn.1972-5787/6232⟩. ⟨hal-01451710⟩

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