  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).
