Notions of anonymous existence in Martin-Löf type theory

As the groupoid model of Hofmann and Streicher proves, identity proofs in intensional Martin-L\"of type theory cannot generally be shown to be unique. Inspired by a theorem by Hedberg, we give some simple characterizations of types that do have unique identity proofs. A key ingredient in these constructions are weakly constant endofunctions on identity types. We study such endofunctions on arbitrary types and show that they always factor through a propositional type, the truncated or squashed domain. Such a factorization is impossible for weakly constant functions in general (a result by Shulman), but we present several non-trivial cases in which it can be done. Based on these results, we define a new notion of anonymous existence in type theory and compare different forms of existence carefully. In addition, we show possibly surprising consequences of the judgmental computation rule of the truncation, in particular in the context of homotopy type theory. All the results have been formalized and verified in the dependently typed programming language Agda

Notions of anonymous existence in Martin-Löf type theory

As the groupoid model of Hofmann and Streicher proves, identity proofs in intensional Martin-L\"of type theory cannot generally be shown to be unique. Inspired by a theorem by Hedberg, we give some simple characterizations of types that do have unique identity proofs. A key ingredient in these constructions are weakly constant endofunctions on identity types. We study such endofunctions on arbitrary types and show that they always factor through a propositional type, the truncated or squashed domain. Such a factorization is impossible for weakly constant functions in general (a result by Shulman), but we present several non-trivial cases in which it can be done. Based on these results, we define a new notion of anonymous existence in type theory and compare different forms of existence carefully. In addition, we show possibly surprising consequences of the judgmental computation rule of the truncation, in particular in the context of homotopy type theory. All the results have been formalized and verified in the dependently typed programming language Agda

Free higher groups in homotopy type theory

Given a type A in homotopy type theory (HoTT), we can define the free ∞-group on A as the higher inductive type F (A)with constructors unit: F(A),cons : A → F(A) → F(A), and conditions saying that every cons(a)is an auto-equivalence on F(A). Equivalently, we can take the loop space of the suspension of A + 1. Assuming that A is a set (i.e. satisfies the principle of unique identity proofs), we are interested in the question whether F(A) is a set as well, which is very much related to an open problem in the HoTT book [20, Ex. 8.2]. We show an approximation to the question, namely that the fundamental groups of F(A) are trivial, i.e. that ∥F(A)∥1 is a set

Domain Theory in Constructive and Predicative Univalent Foundations

We develop domain theory in constructive univalent foundations without Voevodsky's resizing axioms. In previous work in this direction, we constructed the Scott model of PCF and proved its computational adequacy, based on directed complete posets (dcpos). Here we further consider algebraic and continuous dcpos, and construct Scott's $D_\infty$ model of the untyped $\lambda$-calculus. A common approach to deal with size issues in a predicative foundation is to work with information systems or abstract bases or formal topologies rather than dcpos, and approximable relations rather than Scott continuous functions. Here we instead accept that dcpos may be large and work with type universes to account for this. For instance, in the Scott model of PCF, the dcpos have carriers in the second universe $\mathcal{U}_1$ and suprema of directed families with indexing type in the first universe $\mathcal{U}_0$. Seeing a poset as a category in the usual way, we can say that these dcpos are large, but locally small, and have small filtered colimits. In the case of algebraic dcpos, in order to deal with size issues, we proceed mimicking the definition of accessible category. With such a definition, our construction of Scott's $D_\infty$ again gives a large, locally small, algebraic dcpo with small directed suprema.Comment: A shorter version of this paper will appear in the proceedings of CSL 2021, volume 183 of LIPIc

Free higher groups in homotopy type theory

Given a type A in homotopy type theory (HoTT), we can define the free ∞-group on A as the higher inductive type F (A)with constructors unit: F(A),cons : A → F(A) → F(A), and conditions saying that every cons(a)is an auto-equivalence on F(A). Equivalently, we can take the loop space of the suspension of A + 1. Assuming that A is a set (i.e. satisfies the principle of unique identity proofs), we are interested in the question whether F(A) is a set as well, which is very much related to an open problem in the HoTT book [20, Ex. 8.2]. We show an approximation to the question, namely that the fundamental groups of F(A) are trivial, i.e. that ∥F(A)∥1 is a set

Notions of anonymous existence in Martin-Löf type theory

As the groupoid model of Hofmann and Streicher proves, identity proofs in intensional Martin-L\"of type theory cannot generally be shown to be unique. Inspired by a theorem by Hedberg, we give some simple characterizations of types that do have unique identity proofs. A key ingredient in these constructions are weakly constant endofunctions on identity types. We study such endofunctions on arbitrary types and show that they always factor through a propositional type, the truncated or squashed domain. Such a factorization is impossible for weakly constant functions in general (a result by Shulman), but we present several non-trivial cases in which it can be done. Based on these results, we define a new notion of anonymous existence in type theory and compare different forms of existence carefully. In addition, we show possibly surprising consequences of the judgmental computation rule of the truncation, in particular in the context of homotopy type theory. All the results have been formalized and verified in the dependently typed programming language Agda

NOTIONS OF ANONYMOUS EXISTENCE IN MARTIN-LÖF TYPE THEORY

Abstract. As the groupoid model of Hofmann and Streicher proves, identity proofs in intensional Martin-Löf Type Theory can not generally shown to be unique. Inspired by a theorem by Hedberg, we give some simple characterizations of types that do have unique identity proofs. A key ingredient in these constructions are weakly constant endofunctions on identity types. We study such endofunctions on arbitrary types and show that they always factor through a propositional type, the truncated or squashed domain. Followed by a discussion on why this is not possible for weakly constant functions in general, we present certain non-trivial cases in which it can be done. Our results also enable us to define a new notion of anonymous existence in type theory, and different forms of existence are carefully compared. In addition, we show possibly surprising consequences of the judgmental computation rule of the truncation, in particular in the context of homotopy type theory. All the results have been formalized and verified in the dependently typed programming language Agda. 1998 ACM Subject Classification: F.4.1 [Mathematical Logic and Formal Languages]- Lambda calculu