We begin by recalling the essentially global character of universes in various models of
homotopy type theory, which prevents a straightforward axiomatization of their properties …

We define and develop two-level type theory (2LTT), a version of Martin-Löf type theory
which combines two different type theories. We refer to them as the 'inner'and the 'outer'type …

Definitional equality—or conversion—for a type theory with a decidable type checking is the
simplest tool to prove that two objects are the same, letting the system decide just using …

Parametricity is a property of the syntax of type theory implying, eg, that there is only one
function having the type of the polymorphic identity function. Parametricity is usually proven …

Building on the recent extension of dependent type theory with a universe of definitionally
proof-irrelevant types, we introduce TTobs, a new type theory based on the setoidal …

Higher inductive types (HITs) in Homotopy Type Theory allow the definition of datatypes
which have constructors for equalities over the defined type. HITs generalise quotient types …

We propose an abstract notion of a type theory to unify the semantics of various type
theories including Martin–Löf type theory, two-level type theory, and cubical type theory. We …

We present a dependent type theory organized around a Cartesian notion of cubes (with
faces, degeneracies, and diagonals), supporting both fibrant and non-fibrant types. The …

Formal constructive type theory has proved to be an effective language for mechanized
proof. By avoiding non-constructive principles, such as the law of the excluded middle, type …

In this thesis, we study abstract and concrete type theories. We introduce an abstract notion
of a type theory to obtain general results in the semantics of type theories, but we also …