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Homotopy type theory, the Glossary

In mathematical logic and computer science, homotopy type theory (HoTT) refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory applies.^[1]

58 relations: ACM Computing Reviews, André Joyal, ArXiv, Calculus of constructions, Carnegie Mellon University, Categorical logic, Category of sets, Category theory, Coherence condition, Computer science, Coq (software), Creative Commons license, Curry–Howard correspondence, David Corfield, Dependent type, Dimension, ETH Zurich, Fibration, Fork (software development), Formal proof, Foundations of mathematics, Giovanni Felder, GitHub, Groupoid, Higher category theory, Homotopical algebra, Homotopy, Homotopy hypothesis, Homotopy theory, Identity type, Independence (mathematical logic), Institute for Advanced Study, Intuitionistic type theory, Kan fibration, Lambda calculus, Mathematical folklore, Mathematical logic, Michael Shulman (mathematician), Model category, Oberwolfach Research Institute for Mathematics, Path (topology), Path space (algebraic topology), Per Martin-Löf, Peter Aczel, Programming language, Proof assistant, Robert Harper (computer scientist), Simplicial set, Steve Awodey, Structure (mathematical logic), ... Expand index (8 more) »
Foundations of mathematics

ACM Computing Reviews

ACM Computing Reviews (CR) is a scientific journal that reviews literature in the field of computer science.

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André Joyal

André Joyal (born 1943) is a professor of mathematics at the Université du Québec à Montréal who works on category theory.

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ArXiv

arXiv (pronounced as "archive"—the X represents the Greek letter chi ⟨χ⟩) is an open-access repository of electronic preprints and postprints (known as e-prints) approved for posting after moderation, but not peer review.

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Calculus of constructions

In mathematical logic and computer science, the calculus of constructions (CoC) is a type theory created by Thierry Coquand. Homotopy type theory and calculus of constructions are type theory.

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Carnegie Mellon University

Carnegie Mellon University (CMU) is a private research university in Pittsburgh, Pennsylvania.

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Categorical logic

Categorical logic is the branch of mathematics in which tools and concepts from category theory are applied to the study of mathematical logic.

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Category of sets

In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. Homotopy type theory and category of sets are foundations of mathematics.

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Category theory

Category theory is a general theory of mathematical structures and their relations. Homotopy type theory and Category theory are foundations of mathematics.

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Coherence condition

In mathematics, and particularly category theory, a coherence condition is a collection of conditions requiring that various compositions of elementary morphisms are equal.

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Computer science

Computer science is the study of computation, information, and automation.

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Coq (software)

Coq is an interactive theorem prover first released in 1989.

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Creative Commons license

A Creative Commons (CC) license is one of several public copyright licenses that enable the free distribution of an otherwise copyrighted "work".

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Curry–Howard correspondence

In programming language theory and proof theory, the Curry–Howard correspondence (also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions- or formulae-as-types interpretation) is the direct relationship between computer programs and mathematical proofs. Homotopy type theory and Curry–Howard correspondence are type theory.

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David Corfield

David Neil Corfield is a British philosopher specializing in philosophy of mathematics and philosophy of psychology.

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Dependent type

In computer science and logic, a dependent type is a type whose definition depends on a value. Homotopy type theory and dependent type are foundations of mathematics and type theory.

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Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

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ETH Zurich

ETH Zurich (Eidgenössische Technische Hochschule Zürich; Federal Institute of Technology Zurich) is a public research university in Zürich, Switzerland.

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Fibration

The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.

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Fork (software development)

In software engineering, a project fork happens when developers take a copy of source code from one software package and start independent development on it, creating a distinct and separate piece of software.

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Formal proof

In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (known as well-formed formulas when relating to formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence, according to the rule of inference.

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Foundations of mathematics

Foundations of mathematics is the logical and mathematical framework that allows the development of mathematics without generating self-contradictory theories, and, in particular, to have reliable concepts of theorems, proofs, algorithms, etc.

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Giovanni Felder

Giovanni Felder (18 November 1958 in Aarau) is a Swiss mathematical physicist and mathematician, working at ETH Zurich.

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GitHub

GitHub is a developer platform that allows developers to create, store, manage and share their code.

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Groupoid

In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. Homotopy type theory and groupoid are homotopy theory.

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Higher category theory

In mathematics, higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Homotopy type theory and higher category theory are foundations of mathematics.

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Homotopical algebra

In mathematics, homotopical algebra is a collection of concepts comprising the nonabelian aspects of homological algebra, and possibly the abelian aspects as special cases.

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Homotopy

In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from ὁμός "same, similar" and τόπος "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions. Homotopy type theory and homotopy are homotopy theory.

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Homotopy hypothesis

In category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states (very roughly speaking) that the ∞-groupoids are spaces. Homotopy type theory and homotopy hypothesis are homotopy theory.

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Homotopy theory

In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them.

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Identity type

In type theory, the identity type represents the concept of equality. Homotopy type theory and identity type are type theory.

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Independence (mathematical logic)

In mathematical logic, independence is the unprovability of a sentence from other sentences.

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Institute for Advanced Study

The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey.

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Intuitionistic type theory

Intuitionistic type theory (also known as constructive type theory, or Martin-Löf type theory, the latter abbreviated as MLTT) is a type theory and an alternative foundation of mathematics. Homotopy type theory and Intuitionistic type theory are foundations of mathematics and type theory.

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Kan fibration

In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Homotopy type theory and Kan fibration are homotopy theory.

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Lambda calculus

Lambda calculus (also written as λ-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. Homotopy type theory and Lambda calculus are formal methods.

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Mathematical folklore

In common mathematical parlance, a mathematical result is called folklore if it is an unpublished result with no clear originator, but which is well-circulated and believed to be true among the specialists.

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Mathematical logic

Mathematical logic is the study of formal logic within mathematics.

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Michael Shulman (mathematician)

Michael "Mike" Shulman (born 1980) is an American associate professor of mathematics at the University of San Diego who works in category theory and higher category theory, homotopy theory, logic as applied to set theory, and computer science.

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Model category

In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations' satisfying certain axioms relating them. Homotopy type theory and model category are homotopy theory.

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Oberwolfach Research Institute for Mathematics

The Oberwolfach Research Institute for Mathematics (Mathematisches Forschungsinstitut Oberwolfach) is a center for mathematical research in Oberwolfach, Germany.

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Path (topology)

In mathematics, a path in a topological space X is a continuous function from a closed interval into X. Paths play an important role in the fields of topology and mathematical analysis. Homotopy type theory and path (topology) are homotopy theory.

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Path space (algebraic topology)

In algebraic topology, a branch of mathematics, the path space PX of a based space (X, *) is the space that consists of all maps f from the interval I.

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Per Martin-Löf

Per Erik Rutger Martin-Löf (born 8 May 1942) is a Swedish logician, philosopher, and mathematical statistician.

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Peter Aczel

Peter Henry George Aczel (31 October 1941 – 1 August 2023) was a British mathematician, logician and Emeritus joint Professor in the Department of Computer Science and the School of Mathematics at the University of Manchester.

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Programming language

A programming language is a system of notation for writing computer programs.

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Proof assistant

In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human–machine collaboration.

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Robert Harper (computer scientist)

Robert William "Bob" Harper, Jr. (born) is a computer science professor at Carnegie Mellon University who works in programming language research.

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Simplicial set

In mathematics, a simplicial set is an object composed of simplices in a specific way. Homotopy type theory and simplicial set are homotopy theory.

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Steve Awodey

Steven M. Awodey (born 1959) is an American mathematician and logician.

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Structure (mathematical logic)

In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it.

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Thierry Coquand

Thierry Coquand (born 18 April 1961) is a French computer scientist and mathematician who is currently a professor of computer science at the University of Gothenburg, having previously worked at INRIA.

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Thomas Streicher

Thomas Streicher (born 1958) is an Austrian mathematician who is a Professor of Mathematics at Technische Universität Darmstadt.

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Thorsten Altenkirch

Thorsten Altenkirch is a German Professor of Computer Science at the University of Nottingham known for his research on logic, type theory, and homotopy type theory.

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Topology

Topology (from the Greek words, and) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.

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References

[1] https://en.wikipedia.org/wiki/Homotopy_type_theory

Also known as Fibrations-as-Types, Fibrations-as-Types interpretation, Higher inductive type, HoTT, HoTT Book, Homotopic type theory, Homotopical type theory, Mere proposition, Univalence axiom.

, Thierry Coquand, Thomas Streicher, Thorsten Altenkirch, Topology, Type theory, Univalent foundations, Uppsala University, Vladimir Voevodsky.

Homotopy type theory, the Glossary

Table of Contents

See also

Foundations of mathematics

References