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Beck's Monadicity Theorem
In category theory, a branch of mathematics, Beck's monadicity theorem gives a criterion that characterises monadic functors, introduced by in about 1964. It is often stated in dual form for comonads. It is sometimes called the Beck tripleability theorem because of the older term ''triple'' for a monad. Beck's monadicity theorem asserts that a functor :U: C \to D is monadic if and only if # ''U'' has a left adjoint; # ''U'' reflects isomorphisms (if ''U''(''f'') is an isomorphism then so is ''f''); and # ''C'' has coequalizers of ''U''-split parallel pairs (those parallel pairs of morphisms in ''C'', which ''U'' sends to pairs having a split coequalizer in ''D''), and ''U'' preserves those coequalizers. There are several variations of Beck's theorem: if ''U'' has a left adjoint then any of the following conditions ensure that ''U'' is monadic: *''U'' reflects isomorphisms and ''C'' has coequalizers of reflexive pairs (those with a common right inverse) and ''U'' preserves ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism compos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück, and Nigel Hitchin. Currently, the managing editor of Mathematische Annalen is Thomas Schick. Volumes 1–80 (1869–1919) were published by Teubner. Since 1920 (vol. 81), the journal has been published by Springer. In the late 1920s, under the editorship of Hilbert, the journal became embroiled in controversy over the participation of L. E. J. Brouwer on its editorial board, a spillover from the foundational Brouwer–Hilbert controversy. Between 1945 and 1947 the journal briefly ceased publication. References External links''Mathematische Annalen''homepage at Springer''Mathematische Annalen''archive (1869†... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', denotes the result of applying the semigroup operation to the ordered pair . Associativity is formally expressed as that for all ''x'', ''y'' and ''z'' in the semigroup. Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses. The closure axiom is implied by the definition of a binary operation on a set. Some authors thus omit it and specify three axioms for a group and only one axiom (associativity) for a semigroup. As in the case of groups or magmas, the semigroup operation need not be commutative, so ''x''·''y'' is not necessarily equal to ''y''·''x''; a well-known example of an operation that is as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gelfand Duality
In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) is either of two things: * a way of representing commutative Banach algebras as algebras of continuous functions; * the fact that for commutative C*-algebras, this representation is an isometric isomorphism. In the former case, one may regard the Gelfand representation as a far-reaching generalization of the Fourier transform of an integrable function. In the latter case, the Gelfand–Naimark representation theorem is one avenue in the development of spectral theory for normal operators, and generalizes the notion of diagonalizing a normal matrix. Historical remarks One of Gelfand's original applications (and one which historically motivated much of the study of Banach algebras) was to give a much shorter and more conceptual proof of a celebrated lemma of Norbert Wiener (see the citation below), characterizing the elements of the group algebras ''L''1(R) and \ell^1() whose translates ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Ball
Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (album), 1997 album by the Australian band Regurgitator * The Units, a synthpunk band Television * ''The Unit'', an American television series * '' The Unit: Idol Rebooting Project'', South Korean reality TV survival show Business * Stock keeping unit, a discrete inventory management construct * Strategic business unit, a profit center which focuses on product offering and market segment * Unit of account, a monetary unit of measurement * Unit coin, a small coin or medallion (usually military), bearing an organization's insignia or emblem * Work unit, the name given to a place of employment in the People's Republic of China Science and technology Science and medicine * Unit, a vessel or section of a chemical plant * Blood unit, a measuremen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
C*-algebras
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuous linear operators on a complex Hilbert space with two additional properties: * ''A'' is a topologically closed set in the norm topology of operators. * ''A'' is closed under the operation of taking adjoints of operators. Another important class of non-Hilbert C*-algebras includes the algebra C_0(X) of complex-valued continuous functions on ''X'' that vanish at infinity, where ''X'' is a locally compact Hausdorff space. C*-algebras were first considered primarily for their use in quantum mechanics to model algebras of physical observables. This line of research began with Werner Heisenberg's matrix mechanics and in a more mathematically developed form with Pascual Jordan around 1933. Subsequently, John von Neumann attempted to estab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tannakian Category
In mathematics, a Tannakian category is a particular kind of monoidal category ''C'', equipped with some extra structure relative to a given field ''K''. The role of such categories ''C'' is to approximate, in some sense, the category of linear representations of an algebraic group ''G'' defined over ''K''. A number of major applications of the theory have been made, or might be made in pursuit of some of the central conjectures of contemporary algebraic geometry and number theory. The name is taken from Tadao Tannaka and Tannaka–Krein duality, a theory about compact groups ''G'' and their representation theory. The theory was developed first in the school of Alexander Grothendieck. It was later reconsidered by Pierre Deligne, and some simplifications made. The pattern of the theory is that of Grothendieck's Galois theory, which is a theory about finite permutation representations of groups ''G'' which are profinite groups. The gist of the theory, which is rather elaborate in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierre Deligne
Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Prize, and 1978 Fields Medal. Early life and education Deligne was born in Etterbeek, attended school at Athénée Adolphe Max and studied at the Université libre de Bruxelles (ULB), writing a dissertation titled ''Théorème de Lefschetz et critères de dégénérescence de suites spectrales'' (Theorem of Lefschetz and criteria of degeneration of spectral sequences). He completed his doctorate at the University of Paris-Sud in Orsay 1972 under the supervision of Alexander Grothendieck, with a thesis titled ''Théorie de Hodge''. Career Starting in 1972, Deligne worked with Grothendieck at the Institut des Hautes Études Scientifiques (IHÉS) near Paris, initially on the generalization within scheme theory of Zariski's main theorem. In 196 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacques Roubaud
Jacques Roubaud (; born 5 December 1932 in Caluire-et-Cuire, Rhône) is a French poet, writer and mathematician Life and career Jacques Roubaud taught Mathematics at University of Paris X Nanterre and Poetry at EHESS. A member of the Oulipo group, he has published poetry, plays, novels, and translated English poetry and books into French such as Lewis Carroll's ''The Hunting of the Snark''. French poet and novelist Raymond Queneau had Roubaud's first book, a collection of mathematically structured sonnets, published by Éditions Gallimard, and then invited Roubaud to join the Oulipo as the organization's first new member outside the founders.Durand, Marcella"Jacques Roubaud" ''BOMB Magazine''. Summer 2009. Retrieved 26 July 2011. Roubaud's fiction often suppresses the rigorous constraints of the Oulipo (while mentioning their suppression, thereby indicating that such constraints are indeed present), yet takes the Oulipian self-consciousness of the writing act to an extreme. Thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean Bénabou
Jean Bénabou (1932 – 11 February 2022) was a Moroccan-born French mathematician, known for his contributions to category theory. He directed the Research Seminar in Category Theory at the Institut Henri Poincaré and Institut de mathématiques de Jussieu – Paris Rive Gauche, Institut de mathématiques de Jussieu from 1969 to 2001. Bénabou died on 11 February 2022. Partial bibliography * * * * * * See also * References 1932 births 2022 deaths 20th-century French mathematicians Category theorists Moroccan emigrants to France People from Rabat External links * {{France-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
