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Triposes as a generalization of localic geometric morphisms

Published online by Cambridge University Press:  29 January 2021

Jonas Frey  and
Thomas Streicher Open the ORCID record for Thomas Streicher [Opens in a new window]
Jonas Frey
Affiliation:
Department of Philosophy, Carnegie Mellon University, Pittsburgh, PA 15213, USA
Thomas Streicher*
Affiliation:
Fachbereich Mathematik, Technische Universität Darmstadt, Darmstadt, Germany
*
*Corresponding author. Email: streicher@mathematik.tu-darmstadt.de
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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.

Keywords

Fibered categoriestriposestoposes
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 31 , Special Issue 9: In Homage to Martin Hofmann , October 2021 , pp. 1024 - 1033
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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