Related papers: Categorical structures for type theory in univalen…
Category theory unifies mathematical concepts, aiding comparisons across structures by incorporating objects and morphisms, which capture their interactions. It has influenced areas of computer science such as automata theory, functional…
We develop category theory within Univalent Foundations, which is a foundational system for mathematics based on a homotopical interpretation of dependent type theory. In this system, we propose a definition of "category" for which equality…
Algebraic theories with dependency between sorts form the structural core of Martin-L\"of type theory and similar systems. Their denotational semantics are typically studied using categorical techniques; many different categorical…
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…
We show how the categorical logic of untyped, simply typed and dependently typed lambda calculus can be structured around the notion of category with family (cwf). To this end we introduce subcategories of simply typed cwfs (scwfs), where…
We investigate several categories related to transition structures, using a mixture of algebraic and topological methods. We show how two such categories are connected by a contravariant adjunction. This is the most detailed of a family of…
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…
Univalent categories constitute a well-behaved and useful notion of category in univalent foundations. The notion of univalence has subsequently been generalized to bicategories and other structures in (higher) category theory. Here, we…
We present Voevodsky's construction of a model of univalent type theory in the category of simplicial sets. To this end, we first give a general technique for constructing categorical models of dependent type theory, using universes to…
This is an introductory textbook to univalent mathematics and homotopy type theory, a mathematical foundation that takes advantage of the structural nature of mathematical definitions and constructions. It is common in mathematical practice…
We develop bicategory theory in univalent foundations. Guided by the notion of univalence for (1-)categories studied by Ahrens, Kapulkin, and Shulman, we define and study univalent bicategories. To construct examples of univalent…
We present a new coherence theorem for comprehension categories, providing strict models of dependent type theory with all standard constructors, including dependent products, dependent sums, identity types, and other inductive types.…
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…
This paper presents a type theory in which it is possible to directly manipulate $n$-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways…
We investigate two constructive approaches to defining quasi-compact and quasi-separated schemes (qcqs-schemes), namely qcqs-schemes as locally ringed lattices and as functors from rings to sets. We work in Homotopy Type Theory and…
The role of types in categorical models of meaning is investigated. A general scheme for how typed models of meaning may be used to compare sentences, regardless of their grammatical structure is described, and a toy example is used as an…
The Univalence Principle is the statement that equivalent mathematical structures are indistinguishable. We prove a general version of this principle that applies to all set-based, categorical, and higher-categorical structures defined in a…
We study elementary theories of well-pointed toposes and pretoposes, regarded as category-theoretic or "structural" set theories in the spirit of Lawvere's "Elementary Theory of the Category of Sets". We consider weak intuitionistic and…
We construct a univalent universe in the sense of Voevodsky in some suitable model categories for homotopy types (obtained from Grothendieck's theory of test categories). In practice, this means for instance that, appart from the homotopy…
Starting categorically, we give simple and precise models of equivariant classifying spaces. We need these models for work in progress in equivariant infinite loop space theory and equivariant algebraic K-theory, but the models are of…