Related papers: Categories with Dependence and Semantics of Depend…
Seely's paper "Locally cartesian closed categories and type theory" contains a well-known result in categorical type theory: that the category of locally cartesian closed categories is equivalent to the category of Martin-L\"of type…
We study the dependent type theory CaTT, introduced by Finster and Mimram, which presents the theory of weak $\omega$-categories, following the idea that type theories can be considered as presentations of generalized algebraic theories.…
Most categorical models for dependent types have traditionally been heavily set based: contexts form a category, and for each we have a set of types in said context -- and for each type a set of terms of said type. This is the case for…
A non-self-contained gathering of notes on category theory, including the definition of locally cartesian closed category, of the cartesian structure in slice categories, or of the pseudo-cartesian structure on Eilenberg-Moore categories.…
Refinement types are types equipped with predicates that specify preconditions and postconditions of underlying functional languages. We propose a general semantic construction of dependent refinement type systems from underlying type…
A folklore result in category theory is that a (weakly) Cartesian closed category with finite co-products is distributive. Usually, the proof of this small result is carried on using the fact that the exponential functor is right adjoint to…
We build on the correspondence between Petri nets and free symmetric strict monoidal categories already investigated in the literature, and present a categorical semantics for Petri nets with guards. This comes in two flavors: Deterministic…
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…
A relevant category is a symmetric monoidal closed category with a diagonal natural transformation that satisfies some coherence conditions. Every cartesian closed category is a relevant category in this sense. The denomination 'relevant'…
Recently, there has been growing interest in bicategorical models of programming languages, which are "proof-relevant" in the sense that they keep distinct account of execution traces leading to the same observable outcomes, while assigning…
Two novel descriptions of weak {\omega}-categories have been recently proposed, using type-theoretic ideas. The first one is the dependent type theory CaTT whose models are {\omega}-categories. The second is a recursive description of a…
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…
Let $n$ be an integer greater or equal than $3$. We give a simultaneous generalization of $(n-2)$-exact categories and $n$-angulated categories, and we call it one-sided $n$-suspended categories. One-sided $n$-angulated categories are also…
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…
We construct an internal language for cartesian closed bicategories. Precisely, we introduce a type theory modelling the structure of a cartesian closed bicategory and show that its syntactic model satisfies an appropriate universal…
Strong Steiner $\omega$-categories are a class of $\omega$-categories that admit algebraic models in the form of chain complexes, whose formalism allows for several explicit computations. The conditions defining strong Steiner…
The semantics of extensional type theory has an elegant categorical description: models of extensional =-types, 1-types, and Sigma-types are biequivalent to finitely complete categories, while adding Pi-types yields locally Cartesian closed…
In this paper, we define indexed type theories which are related to indexed ($\infty$-)categories in the same way as (homotopy) type theories are related to ($\infty$-)categories. We define several standard constructions for such theories…
We describe all left continuous triangular norms for which the category [0,1]-Cat of real-enriched categories and functors is cartesian closed. We furthermore show that the cartesian closedness of [0,1]-Cat is equivalent to the cartesian…
We introduce $\infty$-type theories as an $\infty$-categorical generalization of the categorical definition of type theories introduced by the second named author. We establish analogous results to the previous work including the…