Internal parametricity, without an interval

TL;DR

This paper introduces a simple extension of Martin-Löf type theory that internalizes parametricity without explicit geometric constructs, using span-based relations and emergent geometry up to dimension 3.

Contribution

It presents a novel type theory with internal parametricity that avoids explicit higher-dimensional cubes and interval sorts, modeled by presheaves over the BCH cube category.

Findings

Modelled by presheaves over BCH cube category

External parametricity and canonicity are proven

Uses span-based rather than relational parametricity

Abstract

Parametricity is a property of the syntax of type theory implying, e.g., that there is only one function having the type of the polymorphic identity function. Parametricity is usually proven externally, and does not hold internally. Internalising it is difficult because once there is a term witnessing parametricity, it also has to be parametric itself and this results in the appearance of higher dimensional cubes. In previous theories with internal parametricity, either an explicit syntax for higher cubes is present or the theory is extended with a new sort for the interval. In this paper we present a type theory with internal parametricity which is a simple extension of Martin-L\"of type theory: there are a few new type formers, term formers and equations. Geometry is not explicit in this syntax, but emergent: the new operations and equations only refer to objects up to dimension 3. We show that this theory is modelled by presheaves over the BCH cube category. Fibrancy conditions are not needed because we use span-based rather than relational parametricity. We define a gluing model for this theory implying that external parametricity and canonicity hold. The theory can be seen as a special case of a new kind of modal type theory, and it is the simplest setting in which the computational properties of higher observational type theory can be demonstrated.