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Homotopy Type Theory
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 as the univalent foundations project. Although neither is precisely delineated, and the terms are sometimes used interchangeably, the choice of usage also sometimes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hott Book Cover
HOTT may refer to: *Mathematics: ** Homotopy type theory *Games: **''Halls of the Things ''Halls of the Things'' is a video game developed by Design Design for the ZX Spectrum and released by Crystal Computing in 1983. It was ported to the Amstrad CPC and Commodore 64. The player travels through seven floors of a tower, searching f ...'', 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] |
Homotopical Algebra
In mathematics, homotopical algebra is a collection of concepts comprising the ''nonabelian'' aspects of homological algebra as well as possibly the abelian aspects as special cases. The ''homotopical'' nomenclature stems from the fact that a common approach to such generalizations is via abstract homotopy theory, as in nonabelian algebraic topology, and in particular the theory of closed model categories. This subject has received much attention in recent years due to new foundational work of Vladimir Voevodsky, Eric Friedlander, Andrei Suslin, and others resulting in the A1 homotopy theory for quasiprojective varieties over a field. Voevodsky has used this new algebraic homotopy theory to prove the Milnor conjecture (for which he was awarded the Fields Medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Type
In type theory, the identity type represents the concept of equality. It is also known as propositional equality to differentiate it from "judgemental equality". Equality in type theory is a complex topic and has been the subject of research, such as the field of homotopy type theory. Comparison with Judgemental Equality The identity type is one of 2 different notions of equality in type theory. The more fundamental notion is "judgemental equality", which is a judgement. Beyond Judgemental Equality The identity type can do more than what judgemental equality can do. It can be used to show "for all x, x+1=1+x", which is impossible to show with judgemental equality. This is accomplished by using the eliminator (or "recursor") of the natural numbers, known as "R". The "R" function let's us define a new function on the natural numbers. That new function "P" is defined to be "(λ x:nat . x+1 = 1+x)". The other arguments act like the parts of an induction proof. The a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Per Martin-Löf
Per Erik Rutger Martin-Löf (; ; born 8 May 1942) is a Swedish logician, philosopher, and 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 judgment, partly inspired by the work of Brentano, Frege, and 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.Member profile |
Simplicial Set
In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. 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. 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 "well-behaved" 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. It is a universal model of computation that can be used to simulate any Turing machine. It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. Lambda calculus consists of constructing § lambda terms and performing § reduction operations on them. In the simplest form of lambda calculus, terms are built using only the following rules: * x – variable, a character or string representing a parameter or mathematical/logical value. * (\lambda x.M) – abstraction, function definition (M is a lambda term). The variable x becomes bound in the expression. * (M\ N) – application, applying a function M to an argument N. M and N are lambda terms. The reduction operations include: * (\lambda x.M \rightarrow(\l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ArXiv
arXiv (pronounced " 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. 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-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, and had hit a million by the end of 2014. As of April 2021, the submission rate is about 16,000 articles per month. History arXiv was made possible by the compact TeX file ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uppsala University
Uppsala University ( sv, Uppsala universitet) is a public research university in Uppsala, Sweden. Founded in 1477, it is the oldest university in Sweden and the Nordic countries still in operation. The university rose to significance during the rise of Sweden as a great power at the end of the 16th century and was then given a relative financial stability with a large donation from King Gustavus Adolphus in the early 17th century. Uppsala also has an important historical place in Swedish national culture, identity and for the Swedish establishment: in historiography, 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 Helsinki. Uppsala belongs to the Coimbra Group of European universities and to the Guild of European Research-Intensive Universities. It has ranked among the wo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. These abstract from the category of topological spaces or of chain complexes (derived category theory). The concept was introduced by . In recent decades, the language of model categories has been used in some parts of algebraic ''K''-theory and algebraic geometry, where homotopy-theoretic approaches led to deep results. Motivation Model categories can provide a natural setting for homotopy theory: the category of topological spaces is a model category, with the homotopy corresponding to the usual theory. Similarly, objects that are thought of as spaces often admit a model category structure, such as the category of simplicial sets. Another model category is the category of chain complexes of ''R''-modules for a commutative ring ''R''. Homotopy th ... [...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), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ... in 2012–13. Bibliography * * References External links * * * * * {{DEFAULTSORT:Awodey, Steve American logicians 20 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Typically the elementary morphisms are part of the data of the category. A coherence theorem states that, in order to be assured that all these equalities hold, it suffices to check a small number of identities. An illustrative example: a monoidal category Part of the data of a monoidal category In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an object ''I'' that is both a left ... is a chosen morphism \alpha_, called the ''associator'': : \alpha_ \colon (A\otimes B)\otimes C \rightarrow A\otimes(B\otimes C) for each triple of objects A, B, C in the category. Using compositions of these \alpha_, one can construct a morphism : ( ( A_N \otimes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
