Notions of parametricity as monoidal models for type theory

TL;DR

This paper develops a categorical framework for understanding parametricity in type theory, showing that cubical models naturally arise as cofree parametric models and providing explicit constructions and examples.

Contribution

It introduces a monoidal model approach to parametricity, generalizes notions of parametricity, and characterizes cubical structures as cofree parametric models within categorical settings.

Findings

Cofree parametric models are explicitly constructed for various categorical structures.

Categories of cubical objects are shown to be cofreely parametric.

Explicit adjoint functors are established between parametric and arbitrary models.

Abstract

This article gives a solid theoretical grounding to the observation that cubical structures arise naturally when working with parametricity. We claim that cubical models are cofreely parametric. We use categories, lex categories or clans as models of type theory. In this context we define notions of parametricity as monoidal models, and parametric models as modules. This covers not only the usual parametricity where any type comes with a relation, but also variants where it comes with a predicate, a reflexive relation, two relations, and many more. In this setting we prove that forgetful functors from parametric models to arbitrary ones have left and right adjoints. Moreover we give explicit compact descriptions for these freely and cofreely parametric models. Then we give many examples of notion of parametricity, allowing to build the following as cofreely parametric models: - Categories of cubical objects for any variant of cube. - Lex categories of truncated semi-cubical (or cubical with reflexivities only) objects. - Clans of Reedy fibrant semi-cubical (or cubical with reflexivities only) objects.