Univalent foundations

About: Univalent foundations is a research topic. Over the lifetime, 95 publications have been published within this topic receiving 4220 citations.

Higher Topos Theory

Abstract: This purpose of this book is twofold: to provide a general introduction to higher category theory (using the formalism of "quasicategories" or "weak Kan complexes"), and to apply this theory to the study of higher versions of Grothendieck topoi. A few applications to classical topology are included.

Homotopy Type Theory: Univalent Foundations of Mathematics

Abstract: Homotopy type theory is a new branch of mathematics that combines aspects of several different fields in a surprising way. It is based on a recently discovered connection between homotopy the- ory and type theory. Homotopy theory is an outgrowth of algebraic topology and homological algebra, with relationships to higher category theory; while type theory is a branch of mathematical logic and theoretical computer science. Although the connections between the two are currently the focus of intense investigation, it is increasingly clear that they are just the beginning of a subject that will take more time and more hard work to fully understand. It touches on topics as seemingly distant as the homotopy groups of spheres, the algorithms for type checking, and the definition of weak ∞-groupoids.

Homotopy theoretic models of identity types

Abstract: Quillen [17] introduced model categories as an abstract framework for homotopy theory which would apply to a wide range of mathematical settings. By all accounts this program has been a success and—as, e.g., the work of Voevodsky on the homotopy theory of schemes [15] or the work of Joyal [11, 12] and Lurie [13] on quasicategories seem to indicate—it will likely continue to facilitate mathematical advances. In this paper we present a novel connection between model categories and mathematical logic, inspired by the groupoid model of (intensional) Martin–Lof type theory [14] due to Hofmann and Streicher [9]. In particular, we show that a form of Martin–Lof type theory can be soundly modelled in any model category. This result indicates moreover that any model category has an associated “internal language” which is itself a form of Martin-Lof type theory. This suggests applications both to type theory and to homotopy theory. Because Martin–Lof type theory is, in one form or another, the theoretical basis for many of the computer proof assistants currently in use, such as Coq and Agda (cf. [3] and [5]), this promise of applications is of a practical, as well as theoretical, nature.

Extensional concepts in intensional type theory

Abstract: The best ebooks about Extensional Concepts In Intensional Type Theory that you can get for free here by download this Extensional Concepts In Intensional Type Theory and save to your desktop. This ebooks is under topic such as extensional concepts in intensional type theory extensional concepts in intensional type theory extensional equality in intensional type theory intensionality, extensionality, and proof irrelevance in how intensional is homotopy type theory? startseite extensional concepts in intensional type theory quotients over minimal type theory linkspringer extensional concepts in intensional type theory dhaze quotients over minimal typetheory unipd about the setoid model csealmers extensional vs intensional logic peregrin quotient types in type theory nottingham eprints a model of type theory in cubical sets naive type theory nottingham quotients over minimal type theory semantic scholar the basics of intensional semantics, part 1: the 3 syntactic properties of propositional equality a simple model for quotient types rdspringer towards intensional/ extensional integration between an approach to the interpretation on intensional contexts a framework for intensional and extensional integration of a model of type theory in cubical sets chalmers interfaces as functors, programs as coalgebras a final de nitional extensions in type theory revisited the syntax and semantics of quantitative type theory intensional-extensional language as etsjets extensional logic of hyperintensions researchgate a primer on homotopy type theory part 1: the formal type quotient completion for the foundation of constructive quotient completion for the foundation of constructive extensional logic of hyperintensions researchgate intensional-extensional language as a measure of semantic intensional semantics for rdf data structures towards an extensional calculus of hyperintensions the calculus of nominal inductive constructions a minimalist two-level foundation for constructive mathematics conceptual description for information modelling based on two lectures on constructive type theory routledge library editions collection oxfordshire quotient types — a modular approach