Cubical Categories for Higher-Dimensional Parametricity

Reynolds' theory of relational parametricity formalizes parametric polymorphism for System F, thus capturing the idea that polymorphically typed System F programs always map related inputs to related results. This paper shows that Reynolds' theory can be seen as the instantiation at dimension 1 of a theory of relational parametricity for System F that holds at all higher dimensions, including infinite dimension. This theory is formulated in terms of the new notion of a p-dimensional cubical category, which we use to define a p-dimensional parametric model of System F for any p, where p is a natural number or infinity. We show that every p-dimensional parametric model of System F yields a split $\lambda$ 2-fibration in which types are interpreted as face map- and degeneracy-preserving cubical functors and terms are interpreted as face map- and degeneracy-preserving cubical natural transformations. We demonstrate that our theory is "good" by showing that the PER model of Bainbridge et al. is derivable as another 1-dimensional instance, and that all instances at all dimensions derive higher-dimensional analogues of expected results for parametric models, such as a Graph Lemma and the existence of initial algebras and final coalgebras. Finally, our technical development resolves a number of significant technical issues arising in Ghani et al.'s recent bifibrational treatment of relational parametricity, which allows us to clarify their approach and strengthen their main result. Once clarified, their bifibrational framework, too, can be seen as a 1-dimensional instance of our theory.