Comprehensive parametric polymorphism: categorical models and type theory

This paper combines reflexive-graph-category structure for relational parametricity with fibrational models of impredicative polymorphism. To achieve this, we modify the definition of fibrational model of impredicative polymorphism by adding one further ingredient to the structure: comprehension in the sense of Lawvere. Our main result is that such comprehensive models, once further endowed with reflexive-graph-category structure, enjoy the expected consequences of parametricity. This is proved using a type-theoretic presentation of the category-theoretic structure, within which the desired consequences of parametricity are derived. The formalisation requires new techniques because equality relations are not available, and standard arguments that exploit equality need to be reworked.