Parametricity and Semi-Cubical Types

TL;DR

This paper constructs a model of type theory that incorporates parametricity by using semi-cubical types, establishing a correspondence between parametricity and cube structures, and extends to generalized algebraic theories.

Contribution

It introduces a method to derive parametric models from arbitrary type theories using semi-cubical types and categorical techniques, broadening the scope of parametricity applications.

Findings

A model of type theory with parametricity is constructed from any given model.

Semi-cubical types serve as the bridge between parametricity and cube structures.

The construction applies to generalized algebraic theories, not just standard type theories.

Abstract

We construct a model of type theory enjoying parametricity from an arbitrary one. A type in the new model is a semi-cubical type in the old one, illustrating the correspondence between parametricity and cubes. Our construction works not only for parametricity, but also for similar interpretations of type theory and in fact similar interpretations of any generalized algebraic theory. To be precise we consider a functor forgetting unary operations and equations defining them recursively in a generalized algebraic theory. We show that it has a right adjoint. We use techniques from locally presentable category theory, as well as from quotient inductive-inductive types.