Cubical Categories for HigherDimensional Parametricity
Abstract
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 pdimensional cubical category, which we use to define a pdimensional parametric model of System F for any p, where p is a natural number or infinity. We show that every pdimensional parametric model of System F yields a split $\lambda$ 2fibration in which types are interpreted as face map and degeneracypreserving cubical functors and terms are interpreted as face map and degeneracypreserving cubical natural transformations. We demonstrate that our theory is "good" by showing that the PER model of Bainbridge et al. is derivable as another 1dimensional instance, and that all instances at all dimensions derive higherdimensional 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 1dimensional instance of our theory.
 Publication:

arXiv eprints
 Pub Date:
 January 2017
 DOI:
 10.48550/arXiv.1701.06244
 arXiv:
 arXiv:1701.06244
 Bibcode:
 2017arXiv170106244J
 Keywords:

 Computer Science  Logic in Computer Science