Related papers: Three non-cubical applications of extension types
We introduce a notion of globular multicategory with homomorphism types. These structures arise when organizing collections of "higher category-like" objects such as type theories with identity types. We show how these globular…
Some derivation-based differential calculi which have been used to construct models of noncommutative gauge theories are presented and commented. Some comparisons between them are made.
We construct a model of type theory enjoying parametricity from an arbitrary one. A type in the new model is a semi-cubical type in the old one, illustrating the correspondence between parametricity and cubes. Our construction works not…
We develop some basic facts on deformations of exterior differential ideals on a smooth complex algebraic variety. With these tools we study deformations of several types of differential ideals, leading to several irreducible components of…
We build free, bigraded bidifferential algebra models for the forms on a complex manifold, with respect to a strong notion of quasi-isomorphism and compatible with the conjugation symmetry. This answers a question of Sullivan. The resulting…
This paper improves the treatment of equality in guarded dependent type theory (GDTT), by combining it with cubical type theory (CTT). GDTT is an extensional type theory with guarded recursive types, which are useful for building models of…
We define a computational type theory combining the contentful equality structure of cartesian cubical type theory with internal parametricity primitives. The combined theory supports both univalence and its relational equivalent, which we…
The representation and the cohomology theory of associative 2-algebras are developed. We study the deformations and abelian extensions of associative 2-algebras in details.
The notion of a duality between two derived functors as well as an extension theorem for derived functors to larger categories in which they need not be defined is introduced. These ideas are then applied to extend and study the coext…
The paper establishes an equivalence between directed homotopy categories of (diagrams of) cubical sets and (diagrams of) directed topological spaces. This equivalence both lifts and extends an equivalence between classical homotopy…
The aim of this article is to explain a philosophy for applying higher dimensional Seifert-van Kampen Theorems, and how the use of groupoids and strict higher groupoids resolves some foundational anomalies in algebraic topology at the…
In this paper we combine the principled approach to modalities from multimodal type theory (MTT) with the computationally well-behaved realization of identity types from cubical type theory (CTT). The result -- cubical modal type theory…
We explore recursive programming with extensible data types. Row types make the structure of data types first class, and can express a variety of type system features including record subtyping and combination of case branches. Our goal is…
The theory of unified product and extending structures for alternative and pre-alternative algebras are developed. It is proved that the extending structures of these algebras can be classified by using some non-abelian cohomology and…
We develop a theory of holomorphic differentials on a certain class of non-compact Riemann surfaces obtained by opening infinitely many nodes.
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…
This paper aims to help the development of new models of homotopy type theory, in particular with models that are based on realizability toposes. For this purpose it develops the foundations of an internal simplicial homotopy that does not…
Cubical type theories are designed around an abstract unit interval from which types of paths, used to represent equalities, are defined. Varying the operations available on this interval yields different type theories. A reversal is an…
There have recently been several developments in synthetic mathematics using extensions of dependent type theory with univalence and higher inductive types: simplicial homotopy type theory, synthetic algebraic geometry and synthetic Stone…
Graph-based signal processing techniques have become essential for handling data in non-Euclidean spaces. However, there is a growing awareness that these graph models might need to be expanded into `higher-order' domains to effectively…