A Model of Parametric Dependent Type Theory in Bridge/Path Cubical Sets

The purpose of this text is to prove all technical aspects of our model for dependent type theory with parametric quantifiers [Nuyts, Vezzosi and Devriese, 2017]. It is well-known that any presheaf category constitutes a model of dependent type theory [Hofmann, 1997], including a hierarchy of universes if the metatheory has one [Hofmann and Streicher, 1997]. We construct our model by defining the base category BPCube of bridge/path cubes and adapting the general presheaf model over BPCube to suit our needs. Our model is heavily based on the models by Atkey, Ghani and Johann [2014], Huber [2015], Bezem, Coquand and Huber [2014], Cohen, Coquand, Huber and M\"ortberg [2016], Moulin [2016] and Bernardy, Coquand and Moulin [2015]. In chapter 1, we review the main concepts of categories with families, and the standard presheaf model of dependent type theory, and we establish the notations we will use. In chapter 2, we capture morphisms of CwFs, and natural transformations and adjunctions between them, in typing rules. We especially study morphisms of CwFs between presheaf categories, that arise from functors between the base categories. In chapter 3, we introduce the category BPCube of bridge/path cubes, and its presheaf category Psh(BPCube) of bridge/path cubical sets. There is a rich interaction with the category of cubical sets Psh(Cube) which we investigate more closely using ideas from axiomatic cohesion [Licata and Shulman, 2016]. In chapter 4, we define discrete types and show that they form a model of dependent type theory. We prove some infrastructural results. In chapter 5, we give an interpretation of the typing rules of ParamDTT [Nuyts, Vezzosi and Devriese, 2017] in Psh(BPCube).