Lawvere-Tierney sheafification in Homotopy Type Theory

Kevin Quirin, Nicolas Tabareau

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 mismatched 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.


Keywords


homotopy type theory; sheaf; modalities

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DOI: 10.6092/issn.1972-5787/6232

Copyright (c) 2016 Kevin Quirin, Nicolas Tabareau

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