Impredicative Encodings of (Higher) Inductive Types

Postulating an impredicative universe in dependent type theory allows System F style encodings of finitary inductive types, but these fail to satisfy the relevant {\eta}-equalities and consequently do not admit dependent eliminators. To recover {\eta} and dependent elimination, we present a method to construct refinements of these impredicative encodings, using ideas from homotopy type theory. We then extend our method to construct impredicative encodings of some higher inductive types, such as 1-truncation and the unit circle S1

A predicative variant of a realizability tripos for the Minimalist Foundation.

open2noHere we present a predicative variant of a realizability tripos validating the intensional level of the Minimalist Foundation extended with Formal Church thesis.the file attached contains the whole number of the journal including the mentioned pubblicationopenMaietti, Maria Emilia; Maschio, SamueleMaietti, MARIA EMILIA; Maschio, Samuel

Categories with New Foundations

While the interaction between set theory and category theory has been studied extensively, the set theories considered have remained almost entirely within the Zermelo family. Quine’s New Foundations has received limited attention, despite being the one-sorted version of a theory mentioned as a possible foundation for Category Theory by Mac Lane and Eilenberg in their seminal paper on the subject. The lack of attention given to NF is not without justification. The category of NF sets is not cartesian closed and the failure of choice is a theorem of NF. But those results should not obscure the aspects of NF that have foundational appeal, nor the value of studying category theory in the context of a universal set. The present research is not intended to “advocate” for the use of NF as a practical foundation for category theory. Instead, the work presents a broad survey of the interaction between the set theory and category theory of NF, examining the relationship in both directions. The abstract structure, of which both type restriction (in the category of NF sets) and size restriction (in the category of all categories) are specific cases, appears to be the study of relative algebra. In a number of cases, the existence of relative algebraic structures in NF can be proven more generally for a class of relative adjoints, (pseudo)monads, etc. Thus, where it seems appropriate to do so, this thesis seeks to contribute to the broader study of relative algebra

Type theory in a type theory with quotient inductive types

Type theory (with dependent types) was introduced by Per Martin-Löf with the intention of providing a foundation for constructive mathematics. A part of constructive mathematics is type theory itself, hence we should be able to say what type theory is using the formal language of type theory. In addition, metatheoretic properties of type theory such as normalisation should be provable in type theory. The usual way of defining type theory formally is by starting with an inductive definition of precontexts, pretypes and preterms and as a second step defining a ternary typing relation over these three components. Well-typed terms are those preterms for which there exists a precontext and pretype such that the relation holds. However, if we use the rich metalanguage of type theory to talk about type theory, we can define well-typed terms directly as an inductive family indexed over contexts and types. We believe that this latter approach is closer to the spirit of type theory where objects come intrinsically with their types. Internalising a type theory with dependent types is challenging because of the mutual definitions of types, terms, substitution of terms and the conversion relation. We use induction induction to express this mutual dependency. Furthermore, to reduce the type-theoretic boilerplate needed for reasoning in the syntax, we encode the conversion relation as the equality type of the syntax. We use equality constructors thus we define the syntax as a quotient inductive type (a special case of higher inductive types from homotopy type theory). We define the syntax of a basic type theory with dependent function space, a base type and a family over the base type as a quotient inductive inductive type. The definition of the syntax comes with a notion of model and an eliminator: whenever one is able to define a model, the eliminator provides a function from the syntax to the model. We show that this method of representing type theory is practically feasible by defining a number of models: the standard model, the logical predicate interpretation for parametricity (as a syntactic translation) and the proof-relevant presheaf logical predicate interpretation. By extending the latter with a quote function back into the syntax, we prove normalisation for type theory. This can be seen as a proof of normalisation by evaluation. Internalising the syntax of type theory is not only of theoretical interest. It opens the possibility of type-theoretic metaprogramming in a type-safe way. This could be used for generic programming in type theory and to implement extensions of type theory which are justified by models such as guarded type theory or homotopy type theory

Type theory in a type theory with quotient inductive types

Type theory (with dependent types) was introduced by Per Martin-Löf with the intention of providing a foundation for constructive mathematics. A part of constructive mathematics is type theory itself, hence we should be able to say what type theory is using the formal language of type theory. In addition, metatheoretic properties of type theory such as normalisation should be provable in type theory. The usual way of defining type theory formally is by starting with an inductive definition of precontexts, pretypes and preterms and as a second step defining a ternary typing relation over these three components. Well-typed terms are those preterms for which there exists a precontext and pretype such that the relation holds. However, if we use the rich metalanguage of type theory to talk about type theory, we can define well-typed terms directly as an inductive family indexed over contexts and types. We believe that this latter approach is closer to the spirit of type theory where objects come intrinsically with their types. Internalising a type theory with dependent types is challenging because of the mutual definitions of types, terms, substitution of terms and the conversion relation. We use induction induction to express this mutual dependency. Furthermore, to reduce the type-theoretic boilerplate needed for reasoning in the syntax, we encode the conversion relation as the equality type of the syntax. We use equality constructors thus we define the syntax as a quotient inductive type (a special case of higher inductive types from homotopy type theory). We define the syntax of a basic type theory with dependent function space, a base type and a family over the base type as a quotient inductive inductive type. The definition of the syntax comes with a notion of model and an eliminator: whenever one is able to define a model, the eliminator provides a function from the syntax to the model. We show that this method of representing type theory is practically feasible by defining a number of models: the standard model, the logical predicate interpretation for parametricity (as a syntactic translation) and the proof-relevant presheaf logical predicate interpretation. By extending the latter with a quote function back into the syntax, we prove normalisation for type theory. This can be seen as a proof of normalisation by evaluation. Internalising the syntax of type theory is not only of theoretical interest. It opens the possibility of type-theoretic metaprogramming in a type-safe way. This could be used for generic programming in type theory and to implement extensions of type theory which are justified by models such as guarded type theory or homotopy type theory

The Creating Subject, the Brouwer-Kripke Schema, and infinite proofs

Kripke's Schema (better the Brouwer-Kripke Schema) and the Kreisel-Troelstra Theory of the Creating Subject were introduced around the same time for the same purpose, that of analysing Brouwer's 'Creating Subject arguments'; other applications have been found since. I first look in detail at a representative choice of Brouwer's arguments. Then I discuss the original use of the Schema and the Theory, their justification from a Brouwerian perspective, and instances of the Schema that can in fact be found in Brouwer's own writings. Finally, I defend the Schema and the Theory against a number of objections that have been made