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An Algebraic Abstraction of the Localic Sheafification via the Tripos-to-Topos Construction
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An Algebraic Abstraction of the Localic Sheafification via the Tripos-to-Topos Construction
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Paper
An Algebraic Abstraction of the Localic Sheafification via the Tripos-to-Topos Construction
2025
Overview
Localic and realizability toposes are two central classes of toposes in categorical logic, both arising through the Hyland-Johnstone-Pitts tripos-to-topos construction. We investigate their shared geometric features by providing an algebraic abstraction of the notions of localic presheaves, sheafification and their connection to supercompactification of a locale via an instance of the Comparison Lemma. This can be applied to a broad class of toposes obtained to the tripos-to-topos constructions, including all those generated from a tripos based on the classical category of ZFC-sets. These results provide a unified geometric framework for understanding localic and realizability toposes.
Publisher
Cornell University Library, arXiv.org
Subject
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