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EXACT COMPLETION AND CONSTRUCTIVE THEORIES OF SETS

Part of: Special categories General theory of categories and functors General logic Algebraic logic Categories with structure Proof theory and constructive mathematics

Published online by Cambridge University Press:  18 June 2020

JACOPO EMMENEGGER  and
ERIK PALMGREN
JACOPO EMMENEGGER
Affiliation:
DEPARTMENT OF MATHEMATICS STOCKHOLM UNIVERSITYSWEDENE-mail: [email protected] address: SCHOOL OF COMPUTER SCIENCE UNIVERSITY OF BIRMINGHAM, UK
ERIK PALMGREN
Affiliation:
DEPARTMENT OF MATHEMATICS STOCKHOLM UNIVERSITYSWEDEN
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Abstract

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In the present paper we use the theory of exact completions to study categorical properties of small setoids in Martin-Löf type theory and, more generally, of models of the Constructive Elementary Theory of the Category of Sets, in terms of properties of their subcategories of choice objects (i.e., objects satisfying the axiom of choice). Because of these intended applications, we deal with categories that lack equalisers and just have weak ones, but whose objects can be regarded as collections of global elements. In this context, we study the internal logic of the categories involved, and employ this analysis to give a sufficient condition for the local cartesian closure of an exact completion. Finally, we apply this result to show when an exact completion produces a model of CETCS.

Keywords

setoidsexact completionlocal cartesian closureconstructive set theorycategorical logic

MSC classification

Primary: 03B15: Higher-order logic and type theory
Secondary: 18B05: Category of sets, characterizations 18D15: Closed categories (closed monoidal and Cartesian closed categories, etc.) 03F55: Intuitionistic mathematics 03G30: Categorical logic, topoi 18A35: Categories admitting limits (complete categories), functors preserving limits, completions
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 2 , June 2020 , pp. 563 - 584
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Footnotes

*

Erik Palmgren passed away in 2019.

References

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