Related papers: Categorical models of subtyping
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,…
Models of dependent type theories are contextual categories with some additional structure. We prove that if a theory $T$ has enough structure, then the category $T\text{-}\mathbf{Mod}$ of its models carries the structure of a model…
This paper presents preliminary work on a general system for integrating dependent types into substructural type systems such as linear logic and linear type theory. Prior work on this front has generally managed to deliver type systems…
We introduce and develop the notion of *displayed categories*. A displayed category over a category C is equivalent to "a category D and functor F : D --> C", but instead of having a single collection of "objects of D" with a map to the…
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…
If M is a model category and Z is an object of M, then there are model category structures on the category of objects of M over Z and the category of objects of M under Z under which a map is a cofibration, fibration, or weak equivalence if…
The notion of a natural model of type theory is defined in terms of that of a representable natural transfomation of presheaves. It is shown that such models agree exactly with the concept of a category with families in the sense of Dybjer,…
The Grothendieck construction is a classical correspondence between diagrams of categories and coCartesian fibrations over the indexing category. In this paper we consider the analogous correspondence in the setting of model categories. As…
Following the types-as-sets paradigm, we present a mechanized embedding of dependent function types with a hierarchy of universes into schematic first-order logic with equality, with axiom schemas of Tarski-Grothendieck set theory. We carry…
In this paper, we define a generalization of indexed categories and contextual categories which we call contextually indexed (contextual) categories. While contextual categories are models of ordinary type theories, contextually indexed…
A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax and complete categorical semantics are…
We introduce the notion of a logical model category which is a Quillen model category satisfying some additional conditions. Those conditions provide enough expressive power that one can soundly interpret dependent products and sums in it.…
The homotopy category of a model structure on a weakly idempotent complete additive category is proved to be equivalent to the additive quotient of the category of cofibrant-fibrant objects with respect to the subcategory of…
Reasoning about weak higher categorical structures constitutes a challenging task, even to the experts. One principal reason is that the language of set theory is not invariant under the weaker notions of equivalence at play, such as…
This paper presents and extends our type theoretical framework for a compositional treatment of natural language semantics with some lexical features like coercions (e.g. of a town into a football club) and copredication (e.g. on a town as…
We define a natural 2-categorical structure on the base category of a large class of Grothendieck fibrations. Given any model category $\mathbf{C}$, we apply this construction to a fibration whose fibers are the homotopy categories of the…
The rules governing the essentially algebraic notion of a category with families have been observed (independently) by Steve Awodey and Marcelo Fiore to precisely match those of a representable natural transformation between presheaves.…
The ability to cast values between related types is a leitmotiv of many flavors of dependent type theory, such as observational type theories, subtyping, or cast calculi for gradual typing. These casts all exhibit a common structural…
A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax and complete categorical semantics are…
We present general techniques for constructing functorial factorizations appropriate for model structures that are not known to be cofibrantly generated. Our methods use "algebraic" characterizations of fibrations to produce factorizations…