Constructing categories and setoids of setoids in type theory

In this paper we consider the problem of building rich categories of setoids, in standard intensional Martin-L\"of type theory (MLTT), and in particular how to handle the problem of equality on objects in this context. Any (proof-irrelevant) family F of setoids over a setoid A gives rise to a category C(A, F) of setoids with objects A. We may regard the family F as a setoid of setoids, and a crucial issue in this article is to construct rich or large enough such families. Depending on closure conditions of F, the category C(A, F) has corresponding categorical constructions. We exemplify this with finite limits. A very large family F may be obtained from Aczel's model construction of CZF in type theory. It is proved that the category so obtained is isomorphic to the internal category of sets in this model. Set theory can thus establish (categorical) properties of C(A, F) which may be used in type theory. We also show that Aczel's model construction may be extended to include the elements of any setoid as atoms or urelements. As a byproduct we obtain a natural extension of CZF, adding atoms. This extension, CZFU, is validated by the extended model. The main theorems of the paper have been checked in the proof assistant Coq which is based on MLTT. A possible application of this development is to integrate set-theoretic and type-theoretic reasoning in proof assistants.Comment: 14 page

On Constructive Sets and Partial Structures

The first three papers in this thesis study the formalisation of a set in type theory as a data type with an equivalence relation – an object usually known as a setoid. The corresponding formalisation of a locally small category is called an E-category. In Paper I, we show that type theory without universes is insufficient for proving that some expected properties hold of the E-category of setoids, but that a minimal universe is sufficient. In Paper II, we show that although the collection of all E-categories does not form a category, we can introduce a type-theoretic version of bicategories, and the E-categories form such an E-bicategory. In Paper III, we consider the setoids inside a type-theoretic universe. The axiom of unique substitutions is proposed and used to show that these form a small category (that is, a category witha setoid of objects and a single setoid of all arrows). We demonstrate that this construction can not be carried out without adding some new axiom to type theory. We also show that the axiom of unique substitutions is strictly weaker than the axiom of unique identity proofs. In Paper IV, we investigate partial equivalence relations, also known as partial setoids, in Heyting arithmetic in all finite types, and adapt the result that the extensional axiom of choice is equivalent to the combination of the intensional axiom of choice, classical logic, and an extensionality axiom. In Paper V, we investigate PHL, a logic of partial terms, and prove a cut elimination theorem for it and for a related calculus

On Constructive Sets and Partial Structures

The first three papers in this thesis study the formalisation of a set in type theory as a data type with an equivalence relation – an object usually known as a setoid. The corresponding formalisation of a locally small category is called an E-category. In Paper I, we show that type theory without universes is insufficient for proving that some expected properties hold of the E-category of setoids, but that a minimal universe is sufficient. In Paper II, we show that although the collection of all E-categories does not form a category, we can introduce a type-theoretic version of bicategories, and the E-categories form such an E-bicategory. In Paper III, we consider the setoids inside a type-theoretic universe. The axiom of unique substitutions is proposed and used to show that these form a small category (that is, a category witha setoid of objects and a single setoid of all arrows). We demonstrate that this construction can not be carried out without adding some new axiom to type theory. We also show that the axiom of unique substitutions is strictly weaker than the axiom of unique identity proofs. In Paper IV, we investigate partial equivalence relations, also known as partial setoids, in Heyting arithmetic in all finite types, and adapt the result that the extensional axiom of choice is equivalent to the combination of the intensional axiom of choice, classical logic, and an extensionality axiom. In Paper V, we investigate PHL, a logic of partial terms, and prove a cut elimination theorem for it and for a related calculus

PERs in HAω I : Basic constructions and choice principles

We study the approximation of sets by partial equivalence rela-tions (PERs) in HAω - this construction is usually considered in type theory.In this rst part, we start with the basic denitions, and introduce notions ofequality, inclusion, and quotients of PERs. We then show that PERs form anitely complete and nitely cocomplete category, with all constructions given.We also make a detailed investigation of extensional choice principles

AN E-BICATEGORY OF E-CATEGORIES EXEMPLIFYING A TYPE-THEORETIC APPROACH TO BICATEGORIES

Abstract. A type-theoretic formalisation of bicategories is introduced, and it is shown that small E-categories, together with their functor categories, form such an E-bicategory. This is carried out using only basic recursive definitions, in the version of predicative type theory with a hierarchy of universes implemented by Agda. This relates to earlier work by Huet and Saïbi, who constructed a large category of small categories in Coq, but with the use of inductive families. The construction presented here may be considered more natural, particularly from the point of view of higher-dimensional category theory. This paper presents a formalisation of some parts of category theory, including a first step towards higher-dimensional category theory. The formalisation is carried out in Agda, a type-theoretic framework with a hierarchy of universes implemented at Chalmers University of Technology, Gothenburg. Agda and Alfa (the version with a graphical interface) are intended to replace the earlier ALF system (see [2, 3, 11]). Further, not all features of the framework were used; restricting mysel

On Constructive Sets and Partial Structures

The first three papers in this thesis study the formalisation of a set in type theory as a data type with an equivalence relation – an object usually known as a setoid. The corresponding formalisation of a locally small category is called an E-category. In Paper I, we show that type theory without universes is insufficient for proving that some expected properties hold of the E-category of setoids, but that a minimal universe is sufficient. In Paper II, we show that although the collection of all E-categories does not form a category, we can introduce a type-theoretic version of bicategories, and the E-categories form such an E-bicategory. In Paper III, we consider the setoids inside a type-theoretic universe. The axiom of unique substitutions is proposed and used to show that these form a small category (that is, a category witha setoid of objects and a single setoid of all arrows). We demonstrate that this construction can not be carried out without adding some new axiom to type theory. We also show that the axiom of unique substitutions is strictly weaker than the axiom of unique identity proofs. In Paper IV, we investigate partial equivalence relations, also known as partial setoids, in Heyting arithmetic in all finite types, and adapt the result that the extensional axiom of choice is equivalent to the combination of the intensional axiom of choice, classical logic, and an extensionality axiom. In Paper V, we investigate PHL, a logic of partial terms, and prove a cut elimination theorem for it and for a related calculus

PERs in HAω I : Basic constructions and choice principles

We study the approximation of sets by partial equivalence rela-tions (PERs) in HAω - this construction is usually considered in type theory.In this rst part, we start with the basic denitions, and introduce notions ofequality, inclusion, and quotients of PERs. We then show that PERs form anitely complete and nitely cocomplete category, with all constructions given.We also make a detailed investigation of extensional choice principles

PERs in HAω I : Basic constructions and choice principles

We study the approximation of sets by partial equivalence rela-tions (PERs) in HAω - this construction is usually considered in type theory.In this rst part, we start with the basic denitions, and introduce notions ofequality, inclusion, and quotients of PERs. We then show that PERs form anitely complete and nitely cocomplete category, with all constructions given.We also make a detailed investigation of extensional choice principles