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Higher Inductive Type
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. This includes, among other lines of work, the construction of homotopical and higher-categorical models for such type theories; the use of type theory as a logic (or internal language) for abstract homotopy theory and higher category theory; the development of mathematics within a type-theoretic foundation (including both previously existing mathematics and new mathematics that homotopical types make possible); and the formalization of each of these in computer proof assistants. There is a large overlap between the work referred to as homotopy type theory, and that called the univalent foundations project. Although neither is precisely delineated, and the terms are sometimes used interchangeably, the choice of usage also so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hott Book Cover
HOTT may refer to: *Mathematics: ** Homotopy type theory *Games: **'' Halls of the Things'', an early video game ** ''Hordes of the Things'' (wargame) *Entertainment: **" Hanging on the Telephone", a song by the power pop band The Nerves, also recorded by Blondie ** Hour of the Time, a shortwave radio show *Other: ** Hot Topic's former NASDAQ ticker symbol {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomas Streicher
Thomas Streicher (11 February 1958 – 2 January 2025) was an Austrian mathematician who was a Professor of Mathematics at Technische Universität Darmstadt. He received his PhD in 1988 from the University of Passau with advisor Manfred Broy. Life and work Streicher's research interests included categorical logic, domain theory and Martin-Löf type theory. In joint work with he constructed a model for intensional type theory, intensional Martin-Löf type theory where identity types are interpreted as groupoids. This was the first model with non-trivial identity types, i.e. other than set (mathematics), sets. Based on this work other models with non-trivial identity types were studied, including homotopy type theory which has been proposed as a foundation for mathematics in Vladimir Voevodsky's research program ''Univalent Foundations of Mathematics''. Together with Martin Hofmann he received the 2014 IEEE Symposium on Logic in Computer Science, LICS Test-of-Time Award for th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Per Martin-Löf
Per Erik Rutger Martin-Löf (; ; born 8 May 1942) is a Sweden, Swedish logician, philosopher, and mathematical statistics, mathematical statistician. He is internationally renowned for his work on the foundations of probability, statistics, mathematical logic, and computer science. Since the late 1970s, Martin-Löf's publications have been mainly in logic. In philosophical logic, Martin-Löf has wrestled with the philosophy of logical consequence and Edmund Husserl#Philosophy of logic and mathematics, judgment, partly inspired by the work of Franz Brentano, Brentano, Gottlob Frege, Frege, and Edmund Husserl, Husserl. In mathematical logic, Martin-Löf has been active in developing intuitionistic type theory as a constructive foundation of mathematics; Martin-Löf's work on type theory has influenced computer science. Until his retirement in 2009, Per Martin-Löf held a joint chair for Mathematics and Philosophy at Stockholm University. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Set
In mathematics, a simplicial set is a sequence of sets with internal order structure ( abstract simplices) and maps between them. Simplicial sets are higher-dimensional generalizations of directed graphs. Every simplicial set gives rise to a "nice" topological space, known as its geometric realization. This realization consists of geometric simplices, glued together according to the rules of the simplicial set. Indeed, one may view a simplicial set as a purely combinatorial construction designed to capture the essence of a topological space for the purposes of homotopy theory. Specifically, the category of simplicial sets carries a natural model structure, and the corresponding homotopy category is equivalent to the familiar homotopy category of topological spaces. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial sets were introduced in 1950 by Samuel Eilenberg and Joseph A. Zilber. Simplic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lambda Calculus
In mathematical logic, the lambda calculus (also written as ''λ''-calculus) is a formal system for expressing computability, computation based on function Abstraction (computer science), abstraction and function application, application using variable Name binding, binding and Substitution (algebra), substitution. Untyped lambda calculus, the topic of this article, is a universal machine, a model of computation that can be used to simulate any Turing machine (and vice versa). It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. In 1936, Church found a formulation which was #History, logically consistent, and documented it in 1940. Lambda calculus consists of constructing #Lambda terms, lambda terms and performing #Reduction, reduction operations on them. A term is defined as any valid lambda calculus expression. In the simplest form of lambda calculus, terms are built using only the following rules: # x: A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ArXiv
arXiv (pronounced as "archive"—the X represents the Chi (letter), 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 Scholarly peer review, peer reviewed. It consists of scientific papers in the fields of mathematics, physics, astronomy, electrical engineering, computer science, quantitative biology, statistics, mathematical finance, and economics, which can be accessed online. In many fields of mathematics and physics, almost all scientific papers are self-archiving, self-archived on the arXiv repository before publication in a peer-reviewed journal. Some publishers also grant permission for authors to archive the peer-reviewed postprint. Begun on August 14, 1991, arXiv.org passed the half-million-article milestone on October 3, 2008, had hit a million by the end of 2014 and two million by the end of 2021. As of November 2024, the submission rate is about 24,000 arti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uppsala University
Uppsala University (UU) () is a public university, public research university in Uppsala, Sweden. Founded in 1477, it is the List of universities in Sweden, oldest university in Sweden and the Nordic countries still in operation. Initially founded in the 15th century, the university rose to significance during the rise of Swedish Empire, Sweden as a great power at the end of the 16th century and was then given relative financial stability with a large donation from Monarchy of Sweden, King Gustavus Adolphus of Sweden, Gustavus Adolphus in the early 17th century. Uppsala also has an important historical place in Swedish national culture, and national identity, identity for the Swedish establishment: in historiography, religion, literature, politics, and music. Many aspects of Swedish academic culture in general, such as the white student cap, originated in Uppsala. It shares some peculiarities, such as the student nation system, with Lund University and the University of Helsink ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Category
A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , . Models can be divided into physical models (e.g. a ship model or a fashion model) and abstract models (e.g. a set of mathematical equations describing the workings of the atmosphere for the purpose of weather forecasting). Abstract or conceptual models are central to philosophy of science. In scholarly research and applied science, a model should not be confused with a theory: while a model seeks only to represent reality with the purpose of better understanding or predicting the world, a theory is more ambitious in that it claims to be an explanation of reality. Types of model ''Model'' in specific contexts As a noun, ''model'' has specific meanings in certain fields, derived from its original meaning of "structural design or layout": * Model (art), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steve Awodey
Steven M. Awodey (; born 1959) is an American mathematician and logician. He is a Professor of Philosophy and Mathematics at Carnegie Mellon University. Biography Awodey studied mathematics and philosophy at the University of Marburg and the University of Chicago. He earned his Ph.D. from Chicago under Saunders Mac Lane in 1997. He is an active researcher in the areas of category theory and logic, and has also written on the philosophy of mathematics. He is one of the originators of the field of homotopy type theory. He was a member of the School of Mathematics at the 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. It has served as the academic home of internationally preeminent scholars, including Albert Ein ... in 2012–13. Bibliography * * References External links * * * * * {{DEFAULTSORT:Awodey, Steve American logicians 20th-cen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Shulman (mathematician)
Michael "Mike" Shulman (; born 1980) is an American 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. Work Shulman did his undergraduate work at the California Institute of Technology and his postgraduate work at the University of Cambridge and the University of Chicago, where he received his Ph.D. in 2009. His doctoral thesis and subsequent work dealt with applications of category theory to homotopy theory. In 2009, he received a National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship. In 2012–13, he was a visiting scholar at the Institute for Advanced Study, where he was one of the official participants in the ''Special Year on Univalent Foundations of Mathematics''. Shulman was one of the principal authors of the book ''Homotopy type theory: Univalent foundations of mathematics'', an informal exposition on the basi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherence Condition
In mathematics, specifically in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or "up to isomorphism". The adjectives such as "pseudo-" and "lax-" are used to refer to the fact equalities are weakened in coherent ways; e.g., pseudo-functor, pseudoalgebra. Coherent isomorphism In some situations, isomorphisms need to be chosen in a coherent way. Often, this can be achieved by choosing canonical isomorphisms. But in some cases, such as prestacks, there can be several canonical isomorphisms and there might not be an obvious choice among them. In practice, coherent isomorphisms arise by weakening equalities; e.g., strict associativity may be replaced by associativity via coherent isomorphisms. For example, via this process, one gets the notion of a weak 2-category from that of a strict 2-category. Replacing coherent isomorphisms by equalities is usually called strictification or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vladimir Voevodsky
Vladimir Alexandrovich Voevodsky (, ; 4 June 1966 – 30 September 2017) was a Russian-American mathematician. His work in developing a homotopy theory for algebraic varieties and formulating motivic cohomology led to the award of a Fields Medal in 2002. He is also known for the proof of the Milnor conjecture and motivic Bloch–Kato conjectures and for the univalent foundations of mathematics and homotopy type theory. Early life and education Vladimir Voevodsky's father, Aleksander Voevodsky, was head of the Laboratory of High Energy Leptons in the Institute for Nuclear Research at the Russian Academy of Sciences. His mother Tatyana was a chemist. Voevodsky attended Moscow State University for a while, but was expelled without a diploma for refusing to attend classes and failing academically. He received his Ph.D. in mathematics from Harvard University in 1992 after being recommended without even applying and without a formal college degree, following several independent p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
