Triposes, exact completions, and Hilbert's ε-operator

Maietti, Maria Emilia; Pasquali, Fabio; Rosolini, Giuseppe

doi:10.1515/tmj-2017-0106

Cited by 18 publications

(54 citation statements)

References 34 publications

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“…Recently, several generalisations of the notion of a Lawvere hyperdoctrine were considered, and we refer for example to [13,15,16] or to [8,24] for higherorder versions. In this work we consider a natural generalisation of the notion of hyperdoctrine, and we call it simply a doctrine.…”

Section: Logical Doctrinesmentioning

confidence: 99%

“…Now we recall from [13,15,26] the notions of existential and universal doctrines, and we refer to [23] for a detailed introduction to the theory of doctrines and hyperdoctrines. For further insights and applications to higher-order logic or realizability, we refer to [8,20,24].…”

Section: Logical Doctrinesmentioning

confidence: 99%

“…The rule appearing in Corollary 20 is called Rule of Choice in [13], while it appears as explicit definability in [19].…”

Section: Logical Principles In Gödel Hyperdoctrinesmentioning

confidence: 99%

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Dialectica Logical Principles

Trotta

Spadetto

Paiva

2021

Logical Foundations of Computer Science

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Gödel' s Dialectica interpretation was designed to obtain a relative consistency proof for Heyting arithmetic, to be used in conjunction with the double negation interpretation to obtain the consistency of Peano arithmetic. In recent years, proof theoretic transformations (socalled proof interpretations) that are based on Gödel's Dialectica interpretation have been used systematically to extract new content from proofs and so the interpretation has found relevant applications in several areas of mathematics and computer science. Following our previous work on 'Gödel fibrations', we present a (hyper)doctrine characterisation of the Dialectica which corresponds exactly to the logical description of the interpretation. To show that we derive the soundness of the interpretation of the implication connective, as expounded on by Spector and Troelstra, in the categorical model. This requires extra logical principles, going beyond intuitionistic logic, namely Markov's Principle (MP) and the Independence of Premise (IP) principle, as well as some choice. We show how these principles are satisfied in the categorical setting, establishing a tight (internal language) correspondence between the logical system and the categorical framework. This tight correspondence should come handy not only when discussing the applications of the Dialectica already known, like its use to extract computational content from (some) classical theorems (proof mining), its use to help to model specific abstract machines, etc. but also to help devise new applications.

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Section: Logical Doctrinesmentioning

confidence: 99%

Section: Logical Doctrinesmentioning

confidence: 99%

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Dialectica Logical Principles

Trotta

Spadetto

Paiva

2021

Logical Foundations of Computer Science

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“…An elementary and existential doctrine P satisfies (RC) if and only if its completion P satisfies (RC) (see [MPR17]).…”

Section: Propositionmentioning

confidence: 99%

“…• by using results in [MPR17] we characterize when an elementary quotient completion is a tripos-to-topos construction.…”

Section: Introductionmentioning

confidence: 99%

Quasi-toposes as elementary quotient completions

Maietti¹,

Pasquali²,

Rosolini³

2021

Preprint

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The elementary quotient completion of an elementary doctrine in the sense of Lawvere was introduced in previous work by the first and third authors. It generalises the exact completion of a category with finite products and weak equalisers. In this paper we characterise when an elementary quotient completion is a quasi-topos. We obtain as a corollary a complete characterisation of when an elementary quotient completions is an elementary topos. As a byproduct we determine also when the elementary quotient completion of a tripos is equivalent to the doctrine obtained via the tripos-to-topos construction.Our results are reminiscent of other works regarding exact completions and put those under a common scheme: in particular, Carboni and Vitale's characterisation of exact completions in terms of their projective objects, Carboni and Rosolini's characterisation of locally cartesian closed exact completions, also in the revision by Emmenegger, and Menni's characterisation of the exact completions which are elementary toposes.The paper contains results presented by the authors at several international meetings in the past years, in particular at Logic Colloquium 2016 and Category Theory 2017, and during the Trimester devoted to Types, Homotopy Type Theory and Verification at the Hausdorff research Institute for Mathematics in 2018. We would like to thank the organisers of these events who gave us the opportunity to present our results.

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On a Generalization of Equilogical Spaces

Pasquali

2018

Log. Univers.

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Triposes, exact completions, and Hilbert's ε-operator

Cited by 18 publications

References 34 publications

Dialectica Logical Principles

Dialectica Logical Principles

Quasi-toposes as elementary quotient completions

On a Generalization of Equilogical Spaces

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