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Eilenberg
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] |
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] |
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. (See e.g. ) In this context it is therefore conventional to write the name without a space. is a topological space with a single nontrivial homotopy group. Let ''G'' be a group and ''n'' a positive integer. A connected topological space ''X'' is called an Eilenberg–MacLane space of type K(G,n), if it has ''n''-th homotopy group \pi_n(X) isomorphic to ''G'' and all other homotopy groups trivial. Assuming that ''G'' is abelian in the case that n > 1, Eilenberg–MacLane spaces of type K(G,n) always exist, and are all weak homotopy equivalent. Thus, one may consider K(G,n) as referring to a weak homotopy equivalence class of spaces. It is common to refer to any representative as "a K(G,n)" or as "a model of K(G,n)". Moreover, it is comm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eilenberg–Steenrod Axioms
In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular homology, developed by Samuel Eilenberg and Norman Steenrod. One can define a homology theory as a sequence of functors satisfying the Eilenberg–Steenrod axioms. The axiomatic approach, which was developed in 1945, allows one to prove results, such as the Mayer–Vietoris sequence, that are common to all homology theories satisfying the axioms. If one omits the dimension axiom (described below), then the remaining axioms define what is called an extraordinary homology theory. Extraordinary cohomology theories first arose in K-theory and cobordism. Formal definition The Eilenberg–Steenrod axioms apply to a sequence of functors H_n from the category of pairs (X,A) of topological spaces to the category of abelian groups, together with a n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Eilenberg
Richard Eilenberg (13 January 1848 – 5 December 1927) was a German composer. Life Born in Merseburg, Eilenberg's musical career began with the study of piano and composition. At 18 years old, he composed his first work, a concert overture. As a volunteer he participated in the Franco-Prussian War from 1870 to 1871. In 1873, Eilenberg became the music director and conductor in Stettin. In 1889, he decided to move to Berlin as a freelance composer, where his second marriage with his wife Dorothee started. They lived on 73 Bremer Street. Eilenberg composed marches and dances for orchestra, harmony and military music, and a ballet ''The Rose of Shiras'', Op. 134. He also composed the operettas ''Comtess Cliquot'' (1909), ''King Midas,'' ''Marietta,'' and ''The Great Prince''. The most notable music that he composed were his marches, including ''The Coronation March'' (for Alexander III of Russia), and ''Janitscharen-Marsch'', Op. 295. Some of his music pieces, attributable to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eilenberg–Moore Algebra
In category theory, a branch of mathematics, a monad is a triple (T, \eta, \mu) consisting of a functor ''T'' from a category to itself and two natural transformations \eta, \mu that satisfy the conditions like associativity. For example, if F, G are functors adjoint functor, adjoint to each other, then T = G \circ F together with \eta, \mu determined by the adjoint relation is a monad. In concise terms, a monad is a Monoid (category theory), monoid in the Category (mathematics), category of endofunctors of some fixed category (an endofunctor is a functor mapping a category to itself). According to John C. Baez, John Baez, a monad can be considered at least in two ways: https://golem.ph.utexas.edu/category/2009/07/the_monads_hurt_my_head_but_no.html # A monad as a generalized monoid; this is clear since a monad is a monoid in a certain category, # A monad as a tool for studying Algebraic gadget, algebraic gadgets; for example, a group (mathematics), group can be described by a c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
