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Triposes as a generalization of localic geometric morphisms

Triposes as a generalization of localic geometric morphisms Abstract In Hyland et al. (1980), Hyland, Johnstone and Pitts introduced the notion of tripos for the purpose of organizing the construction of realizability toposes in a way that generalizes the construction of localic toposes from complete Heyting algebras. In Pitts (2002), one finds a generalization of this notion eliminating an unnecessary assumption of Hyland et al. (1980). The aim of this paper is to characterize triposes over a base topos ${\cal S}$ in terms of so-called constant objects functors from ${\cal S}$ to some elementary topos. Our characterization is slightly different from the one in Pitts’s PhD Thesis (Pitts, 1981) and motivated by the fibered view of geometric morphisms as described in Streicher (2020). In particular, we discuss the question whether triposes over Set giving rise to equivalent toposes are already equivalent as triposes. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematical Structures in Computer Science Cambridge University Press

Triposes as a generalization of localic geometric morphisms

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Publisher
Cambridge University Press
Copyright
© The Author(s), 2021. Published by Cambridge University Press
ISSN
1469-8072
eISSN
0960-1295
DOI
10.1017/S0960129520000304
Publisher site
See Article on Publisher Site

Abstract

Abstract In Hyland et al. (1980), Hyland, Johnstone and Pitts introduced the notion of tripos for the purpose of organizing the construction of realizability toposes in a way that generalizes the construction of localic toposes from complete Heyting algebras. In Pitts (2002), one finds a generalization of this notion eliminating an unnecessary assumption of Hyland et al. (1980). The aim of this paper is to characterize triposes over a base topos ${\cal S}$ in terms of so-called constant objects functors from ${\cal S}$ to some elementary topos. Our characterization is slightly different from the one in Pitts’s PhD Thesis (Pitts, 1981) and motivated by the fibered view of geometric morphisms as described in Streicher (2020). In particular, we discuss the question whether triposes over Set giving rise to equivalent toposes are already equivalent as triposes.

Journal

Mathematical Structures in Computer ScienceCambridge University Press

Published: Oct 1, 2021

Keywords: Fibered categories; triposes; toposes

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