Dialectica models of type theory

Moss, Sean K.; Glehn, Tamara von

doi:10.1145/3209108.3209207

Cited by 10 publications

(9 citation statements)

References 37 publications

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“…Suppose given two morphisms L and R in the 2-category K//K as in (12). In that case, It should be mentioned that a similar observation is made by Kelly [9] (Prop.…”

Section: Formal Adjunctions In the 2-category K//kmentioning

confidence: 61%

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Comprehension and Quotient Structures in the Language of 2-Categories

Melliès,

Rolland

2020

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Lawvere observed in his celebrated work on hyperdoctrines that the set-theoretic schema of comprehension can be elegantly expressed in the functorial language of categorical logic, as a comprehension structure on the functor p ∶ E → B defining the hyperdoctrine. In this paper, we formulate and study a strictly ordered hierarchy of three notions of comprehension structure on a given functor p ∶ E → B, which we call (i) comprehension structure, (ii) comprehension structure with section, and (iii) comprehension structure with image. Our approach is 2-categorical and we thus formulate the three levels of comprehension structure on a general morphism p ∶ E → B in a 2-category K. This conceptual point of view on comprehension structures enables us to revisit the work by Fumex, Ghani and Johann on the duality between comprehension structures and quotient structures on a given functor p ∶ E → B. In particular, we show how to lift the comprehension and quotient structures on a functor p ∶ E → B to the categories of algebras or coalgebras associated to functors F E ∶ E → E and F B ∶ B → B of interest, in order to interpret reasoning by induction and coinduction in the traditional language of categorical logic, formulated in an appropriate 2-categorical way.

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“…Suppose given two morphisms L and R in the 2-category K//K as in (12). In that case, It should be mentioned that a similar observation is made by Kelly [9] (Prop.…”

Section: Formal Adjunctions In the 2-category K//kmentioning

confidence: 61%

“…Ehrhard's D-categories are also called comprehension category with units by Jacobs [6] def. 4.12, and Ehrhard comprehension category by Moss [12], p 22. A pre-D-category is defined in [4] (Section 2.1, def.…”

Section: Illustration: Inductive Reasoning On Functor Algebras and Du...mentioning

confidence: 99%

Comprehension and Quotient Structures in the Language of 2-Categories

Melliès,

Rolland

2020

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“…But it also opens new possibilities for modelling of constructive set theories (in the style of Nemoto and Rathjan [19]) and of categorical modelling of intermediate logics (intuitionistic propositional logic plus (IP) or (MK), see [1,6]). This leads into applications both into the investigation of functional abstract machines [18,22], of reverse mathematics [19] and of quantified modal logic [25].…”

Section: Discussionmentioning

confidence: 99%

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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“…The notions of right-weak and left-weak adjunction appear in a similar context in [19]. It turns out that the property of existence of right-weakly left adjoints to the cartesian liftings of the injections is enough to make the simple coproduct completion of fibrations preserve the existence of fibred weak finite coproducts.…”

Section: On Fibred (Weak) Finite (Co)productsmentioning

confidence: 99%

“…We recall that the notions of right-weak and left-weak adjunction appear in a similar context in [19].…”

Section: A)mentioning

confidence: 99%

The Gödel Fibration

Trotta,

Spadetto,

de Paiva

2021

Preprint

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We introduce the notion of a Gödel fibration, which is a fibration categorically embodying both the logical principle of traditional Skolemization (we can exchange the order of quantifiers paying the price of a functional) and the existence of a prenex normal form presentation for every logical formula. Building up from Hofstra's earlier fibrational characterization of the de Paiva's categorical Dialectica construction, we show that a fibration is an instance of the Dialectica construction if and only if it is a Gödel fibration. This result establishes an internal presentation of the Dialectica construction. Then we provide a deep structural analysis of the Dialectica construction producing a full description of which categorical structure behaves well with respect to this construction, focusing on (weak) finite products and coproducts. We conclude describing the applications we envisage for this generalized fibrational version of the Dialectica construction. * β ) making the usual triangles commute and, if π I , π X * α × π I , π Y * β is strict, it is the unique one by the universal properties of X ×Y and π I , π X * α× π I , π Y * β in B and E I×X×Y respectively. Moreover observe that, whenever J h − → I is an arrow of B, it is the case that the h-reindexing of ( I, X × Y, π I , π X * α × π I , π Y * β ) is the object:

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Dialectica models of type theory

Cited by 10 publications

References 37 publications

Comprehension and Quotient Structures in the Language of 2-Categories

Comprehension and Quotient Structures in the Language of 2-Categories

Dialectica Logical Principles

The Gödel Fibration

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