Related papers: A Model of Parametric Dependent Type Theory in Bri…
Categories with families (CwFs) have been used to define the semantics of type theory in type theory. In the setting of Homotopy Type Theory (HoTT), one of the limitations of the traditional notion of CwFs is the requirement to set-truncate…
In order to avoid well-know paradoxes associated with self-referential definitions, higher-order dependent type theories stratify the theory using a countably infinite hierarchy of universes (also known as sorts), Type$_0$ : Type$_1$ :…
We develop normalisation by evaluation (NBE) for dependent types based on presheaf categories. Our construction is formulated in the metalanguage of type theory using quotient inductive types. We use a typed presentation hence there are no…
Parametricity is a key metatheoretic property of type systems, which implies strong uniformity & modularity properties of the structure of types within systems possessing it. In recent years, various systems of dependent type theory have…
Dependently typed proof assistant rely crucially on definitional equality, which relates types and terms that are automatically identified in the underlying type theory. This paper extends type theory with definitional functor laws,…
We introduce basic notions in category theory to type theorists, including comprehension categories, categories with attributes, contextual categories, type categories, and categories with families along with additional discussions that are…
We define a general class of dependent type theories, encompassing Martin-L\"of's intuitionistic type theories and variants and extensions. The primary aim is pragmatic: to unify and organise their study, allowing results and constructions…
For a dependent theory T, in C_T for every type definable group G, the intersection of type definable subgroups with bounded index is a type definable subgroup with bounded index.
Structural independence is the (conditional) independence that arises from the structure rather than the precise numerical values of a distribution. We develop this concept and relate it to $d$-separation and structural causal models.…
In the pure Calculus of Constructions (CC) one can define data types and function over these, and there is a powerful higher order logic to reason over these functions and data types. This is due to the combination of impredicativity and…
We exhibit a computational type theory which combines the higher-dimensional structure of cartesian cubical type theory with the internal parametricity primitives of parametric type theory, drawing out the similarities and distinctions…
We develop a dependent type theory that is based purely on inductive and coinductive types, and the corresponding recursion and corecursion principles. This results in a type theory with a small set of rules, while still being fairly…
This article gives a solid theoretical grounding to the observation that cubical structures arise naturally when working with parametricity. We claim that cubical models are cofreely parametric. We use categories, lex categories or clans as…
We extend Levy's call-by-push-value (CBPV) analysis from simple to dependent type theory (DTT) in order to study the interaction between computational effects and dependent types. We define the naive system of dependently typed CBPV,…
This thesis captures the ongoing development of twisted cubes, which is a modification of cubes (in a topological sense) where its homotopy type theory does not require paths or higher paths to be invertible. My original motivation to…
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 using the internal language of the presheaf toposes from…
This paper is the third paper of a series devoted to higher dimensional transition systems. The preceding paper proved the existence of a left determined model structure on the category of cubical transition systems. In this sequel, it is…
This elementary survey article was prepared for a talk at the 2016 Superschool on Derived Categories and D-branes. The goal is to outline an identification of the bounded derived category of coherent sheaves on a Calabi-Yau threefold with…
In dependent type theory, being able to refer to a type universe as a term itself increases its expressive power, but requires mechanisms in place to prevent Girard's paradox from introducing logical inconsistency in the presence of…
Graded type theories are an emerging paradigm for augmenting the reasoning power of types with parameterizable, fine-grained analyses of program properties. There have been many such theories in recent years which equip a type theory with…