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Samuel Eilenberg
Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra. Early life and education He was born in Warsaw, Kingdom of Poland to a Jewish family. He spent much of his career as a professor at Columbia University. He earned his Ph.D. from University of Warsaw in 1936, with thesis ''On the Topological Applications of Maps onto a Circle''; his thesis advisors were Kazimierz Kuratowski and Karol Borsuk. He died in New York City in January 1998. Career Eilenberg's main body of work was in algebraic topology. He worked on the axiomatic treatment of homology theory with Norman Steenrod (and the Eilenberg–Steenrod axioms are named for the pair), and on homological algebra with Saunders Mac Lane. As a result of this work, Eilenberg and Mac Lane developed the field of category theory, for which they are now best known. Eilenberg was a member of Bourbaki and, with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Warsaw
Warsaw, officially the Capital City of Warsaw, is the capital and List of cities and towns in Poland, largest city of Poland. The metropolis stands on the Vistula, River Vistula in east-central Poland. Its population is officially estimated at 1.86 million residents within a Warsaw metropolitan area, greater metropolitan area of 3.27 million residents, which makes Warsaw the List of cities in the European Union by population within city limits, 6th most-populous city in the European Union. The city area measures and comprises List of districts and neighbourhoods of Warsaw, 18 districts, while the metropolitan area covers . Warsaw is classified as an Globalization and World Cities Research Network#Alpha 2, alpha global city, a major political, economic and cultural hub, and the country's seat of government. It is also the capital of the Masovian Voivodeship. Warsaw traces its origins to a small fishing town in Masovia. The city rose to prominence in the late 16th cent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Lawvere
Francis William Lawvere (; February 9, 1937 – January 23, 2023) was an American mathematician known for his work in category theory, topos theory and the philosophy of mathematics. Biography Born in Muncie, Indiana, and raised on a farm outside Mathews, Lawvere received his undergraduate degree in mathematics from Indiana University. Lawvere studied continuum mechanics and kinematics as an undergraduate with Clifford Truesdell. He learned of category theory while teaching a course on functional analysis for Truesdell, specifically from a problem in John L. Kelley's textbook ''General Topology''. Lawvere found it a promising framework for simple rigorous axioms for the physical ideas of Truesdell and Walter Noll. Truesdell supported Lawvere's application to study further with Samuel Eilenberg, a founder of category theory, at Columbia University in 1960. Before completing the Ph.D. Lawvere spent a year in Berkeley as an informal student of model theory and set theory, foll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eilenberg–Ganea Conjecture
The Eilenberg–Ganea conjecture is a claim in algebraic topology. It was formulated by Samuel Eilenberg and Tudor Ganea in 1957, in a short, but influential paper. It states that if a group ''G'' has cohomological dimension 2, then it has a 2-dimensional Eilenberg–MacLane space In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. ... K(G,1). For ''n'' different from 2, a group ''G'' of cohomological dimension ''n'' has an ''n''-dimensional Eilenberg–MacLane space. It is also known that a group of cohomological dimension 2 has a 3-dimensional Eilenberg−MacLane space. In 1997, Mladen Bestvina and Noel Brady constructed a group ''G'' so that either ''G'' is a counterexample to the Eilenberg–Ganea conjecture, or there must be a counterexample to the Whitehead conjectur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eilenberg Swindle
Eilenberg is a surname. Notable people with the surname include: * Samuel Eilenberg (1913–1998), Polish mathematician * Richard Eilenberg (1848–1927), German composer Named after Samuel * Eilenberg–MacLane space * Eilenberg–Moore algebra * Eilenberg–Steenrod axioms * Eilenberg machine See also * Eilenburg Eilenburg (; , ) is a town in Germany. It lies in the district of Nordsachsen in Saxony, approximately 20 km northeast of the city of Leipzig. Geography Eilenburg lies at the banks of the river Mulde at the southwestern edge of the D� ... * Eulenberg (other) {{surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Obstruction Theory
Obstruction may refer to: Places * Obstruction Island, in Washington state * Obstruction Islands, east of New Guinea Medicine * Obstructive jaundice * Obstructive sleep apnea * Airway obstruction, a respiratory problem ** Recurrent airway obstruction * Bowel obstruction, a blockage of the intestines. * Gastric outlet obstruction * Distal intestinal obstruction syndrome * Congenital lacrimal duct obstruction * Bladder outlet obstruction Politics and law * Obstruction of justice, the crime of interfering with law enforcement * Obstructionism, the practice of deliberately delaying or preventing a process or change, especially in politics * Emergency Workers (Obstruction) Act 2006 Science and mathematics * Obstruction set in forbidden graph characterizations, in the study of graph minors in graph theory * Obstruction theory, in mathematics * Propagation path obstruction ** Single Vegetative Obstruction Model Sports * Obstruction (baseball), when a fielder illegally hinders a baser ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standard Complex
In mathematics, the bar complex, also called the bar resolution, bar construction, standard resolution, or standard complex, is a way of constructing resolutions in homological algebra. It was first introduced for the special case of algebras over a commutative ring by and and has since been generalized in many ways. The name "bar complex" comes from the fact that used a vertical bar , as a shortened form of the tensor product \otimes in their notation for the complex. Definition Let R be an algebra over a field k, let M_1 be a right R- module, and let M_2 be a left R-module. Then, one can form the bar complex \operatorname_R(M_1,M_2) given by :\cdots\rightarrow M_1 \otimes_k R \otimes_k R \otimes_k M_2 \rightarrow M_1 \otimes_k R \otimes_k M_2 \rightarrow M_1 \otimes_k M_2 \rightarrow 0\,, with the differential :\begin d(m_1 \otimes r_1 \otimes \cdots \otimes r_n \otimes m_2) &= m_1 r_1 \otimes \cdots \otimes r_n \otimes m_2 \\ &+ \sum_^ (-1)^i m_1 \otimes r_1 \otimes \cdo ... [...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] |
Shuffle Algebra
In mathematics, a shuffle algebra is a Hopf algebra with a basis corresponding to words on some set, whose product is given by the shuffle product ''X'' ⧢ ''Y'' of two words ''X'', ''Y'': the sum of all ways of interlacing them. The interlacing is given by the riffle shuffle permutation. The shuffle algebra on a finite set is the graded dual of the universal enveloping algebra of the free Lie algebra on the set. Over the rational numbers, the shuffle algebra is isomorphic to the polynomial algebra in the Lyndon words. The shuffle product occurs in generic settings in non-commutative algebras; this is because it is able to preserve the relative order of factors being multiplied together - the riffle shuffle permutation. This can be held in contrast to the divided power structure, which becomes appropriate when factors are commutative. Shuffle product The shuffle product of words of lengths ''m'' and ''n'' is a sum over the ways of interleaving the two words, as shown in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Module
In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below. Every free module is a projective module, but the converse fails to hold over some rings, such as Dedekind rings that are not principal ideal domains. However, every projective module is a free module if the ring is a principal ideal domain such as the integers, or a (multivariate) polynomial ring over a field (this is the Quillen–Suslin theorem). Projective modules were first introduced in 1956 in the influential book ''Homological Algebra'' by Henri Cartan and Samuel Eilenberg. Definitions Lifting property The usual category theoretical definition is in terms of the property of ''lifting'' that carries over from free to projective modules: a module ''P'' is projective if and only if fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Dimension
In abstract algebra, the weak dimension of a nonzero right module ''M'' over a ring ''R'' is the largest number ''n'' such that the Tor group \operatorname_n^R(M,N) is nonzero for some left ''R''-module ''N'' (or infinity if no largest such ''n'' exists), and the weak dimension of a left ''R''-module is defined similarly. The weak dimension was introduced by . The weak dimension is sometimes called the flat dimension as it is the shortest length of the resolution of the module by flat modules. The weak dimension of a module is, at most, equal to its projective dimension. The weak global dimension of a ring is the largest number ''n'' such that \operatorname_n^R(M,N) is nonzero for some right ''R''-module ''M'' and left ''R''-module ''N''. If there is no such largest number ''n'', the weak global dimension is defined to be infinite. It is at most equal to the left or right global dimension of the ring ''R''. Examples *The module \Q of rational numbers over the ring \Z of integ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
X-machine
The X-machine (''XM'') is a theoretical model of computation introduced by Samuel Eilenberg in 1974.S. Eilenberg (1974) ''Automata, Languages and Machines, Vol. A''. Academic Press, London. The ''X'' in "X-machine" represents the fundamental data type on which the machine operates; for example, a machine that operates on databases (objects of type ''database'') would be a ''database''-machine. The X-machine model is structurally the same as the finite-state machine, except that the symbols used to label the machine's transitions denote relations of type ''X''→''X''. Crossing a transition is equivalent to applying the relation that labels it (computing a set of changes to the data type ''X''), and traversing a path in the machine corresponds to applying all the associated relations, one after the other. Original theory Eilenberg's original X-machine was a completely general theoretical model of computation (subsuming the Turing machine, for example), which admitted determini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in most areas of mathematics. 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 space (other), quotient spaces, direct products, completion, and duality (mathematics), duality. Many areas of computer science also rely on category theory, such as functional programming and Semantics (computer science), semantics. A category (mathematics), category is formed by two sorts of mathematical object, objects: the object (category theory), objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. Metapho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
