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Daniel Kan
Daniel Marinus Kan (or simply Dan Kan) (August 4, 1927 – August 4, 2013) was a Dutch mathematician working in category theory and homotopy theory. He was a prolific contributor to both fields for six decades, having authored or coauthored several dozen research papers and monographs. Career He received his Ph.D. at Hebrew University in 1955, under the direction of Samuel Eilenberg. His students include Aldridge K. Bousfield, William Dwyer, Stewart Priddy, Emmanuel Dror Farjoun and Jeffrey H. Smith. He was an emeritus professor at the Massachusetts Institute of Technology where he taught from 1959, formally retiring in 1993. Work He played a role in the beginnings of modern homotopy theory similar to that of Saunders Mac Lane in homological algebra, namely the adroit and persistent application of categorical methods. His most famous work is the abstract formulation of the discovery of adjoint functors, which dates from 1958. The Kan extension is one of the broadest descript ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kan-Thurston Theorem
In mathematics, particularly algebraic topology, the Kan-Thurston theorem associates a discrete group G to every path-connected topological space X in such a way that the group cohomology of G is the same as the cohomology of the space X. The group G might then be regarded as a good approximation to the space X, and consequently the theorem is sometimes interpreted to mean that homotopy theory can be viewed as part of group theory. More precisely, the theorem states that every path-connected topological space is homology-equivalent to the classifying space K(G,1) of a discrete group G, where homology-equivalent means there is a map K(G,1) \rightarrow X inducing an isomorphism on homology. The theorem is attributed to Daniel Kan and William Thurston who published their result in 1976. Statement of the Kan-Thurston theorem Let X be a path-connected topological space. Then, naturally associated to X, there is a Serre fibration t_x \colon T_X \to X where T_X is an aspherica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Amsterdam
Amsterdam ( , , , lit. ''The Dam on the River Amstel'') is the Capital of the Netherlands, capital and Municipalities of the Netherlands, most populous city of the Netherlands, with The Hague being the seat of government. It has a population of 907,976 within the city proper, 1,558,755 in the City Region of Amsterdam, urban area and 2,480,394 in the Amsterdam metropolitan area, metropolitan area. Located in the Provinces of the Netherlands, Dutch province of North Holland, Amsterdam is colloquially referred to as the "Venice of the North", for its large number of canals, now designated a World Heritage Site, UNESCO World Heritage Site. Amsterdam was founded at the mouth of the Amstel River that was dammed to control flooding; the city's name derives from the Amstel dam. Originally a small fishing village in the late 12th century, Amsterdam became a major world port during the Dutch Golden Age of the 17th century, when the Netherlands was an economic powerhouse. Amsterdam is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stewart Priddy
Stewart may refer to: People *Stewart (name), Scottish surname and given name *Clan Stewart, a Scottish clan * Clan Stewart of Appin, a Scottish clan Places Canada *Stewart, British Columbia *Stewart Township, Nipissing District, Ontario (historical) New Zealand * Stewart Island / Rakiura United Kingdom * Newton Stewart, Scotland * Portstewart, Northern Ireland *Stewartby, Bedfordshire, England United States Airports *Stewart Air Force Base, New York, a former Air Force base and now-joint civil-military airport, shared by: ** Stewart Air National Guard Base, New York ** Stewart International Airport (also known as Newburgh-Stewart IAP), New York Counties * Stewart County, Georgia * Stewart County, Tennessee Localities *Stewart, Alabama *Stewart, Indiana *Stewart, Minnesota *Stewart, Mississippi *Stewart, Missouri *Stewart, Ohio * Stewart, Tennessee *Stewart, Texas *Stewart, West Virginia *Fort Stewart, Georgia *Stewart Manor, New York, a village in the Town of Hempstead, in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hebrew University Of Jerusalem Alumni
Hebrew (; ; ) is a Northwest Semitic language of the Afroasiatic language family. Historically, it is one of the spoken languages of the Israelites and their longest-surviving descendants, the Jews and Samaritans. It was largely preserved throughout history as the main liturgical language of Judaism (since the Second Temple period) and Samaritanism. Hebrew is the only Canaanite language still spoken today, and serves as the only truly successful example of a dead language that has been revived. It is also one of only two Northwest Semitic languages still in use, with the other being Aramaic. The earliest examples of written Paleo-Hebrew date back to the 10th century BCE. Nearly all of the Hebrew Bible is written in Biblical Hebrew, with much of its present form in the dialect that scholars believe flourished around the 6th century BCE, during the time of the Babylonian captivity. For this reason, Hebrew has been referred to by Jews as '' Lashon Hakodesh'' (, ) since anci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Israeli Expatriates In The United States
Israeli may refer to: * Something of, from, or related to the State of Israel * Israelis, citizens or permanent residents of the State of Israel * Modern Hebrew, a language * ''Israeli'' (newspaper), published from 2006 to 2008 * Guni Israeli (born 1984), Israeli basketball player See also * Israelites, the ancient people of the Land of Israel * List of Israelis Israelis ( he, ישראלים ''Yiśraʾelim'') are the citizens or permanent residents of the State of Israel, a multiethnic state populated by people of different ethnic backgrounds. The largest ethnic groups in Israel are Jews (75%), foll ... {{disambiguation Language and nationality disambiguation pages ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Colimit
In mathematics, especially in algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category \text(\textbf). The main idea is this: if we have a diagramF: I \to \textbfconsidered as an object in the homotopy category of diagrams F \in \text(\textbf^I), (where the homotopy equivalence of diagrams is considered pointwise), then the homotopy limit and colimits then correspond to the cone and cocone\begin \underset(F)&: * \to \textbf\\ \underset(F)&: * \to \textbf \endwhich are objects in the homotopy category \text(\textbf^*), where * is the category with one object and one morphism. Note this category is equivalent to the standard homotopy category \text(\textbf) since the latter homotopy functor category has functors which picks out an object in \text and a natural transformation corresponds to a continuous function of topological spaces. Note this construction can be generalized to model categories, which g ... [...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] |
Kan Complex
In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category structure on simplicial sets and are therefore of fundamental importance. Kan complexes are the fibrant objects in this model category. The name is in honor of Daniel Kan. Definitions Definition of the standard n-simplex For each ''n'' ≥ 0, recall that the standard n-simplex, \Delta^n, is the representable simplicial set :\Delta^n(i) = \mathrm_ ( Applying the geometric realization functor to this simplicial set gives a space homeomorphic to the topological standard n-simplex: the convex subspace of ℝn+1 consisting of all points (t_0,\dots,t_n) such that the coordinates are non-negative and sum to 1. Definition of a horn For each ''k'' ≤ ''n'', this has a subcomplex \Lambda^n_k, the ''k''-th horn inside \Delta^n, corresponding to the boundary of the ''n''-simplex, with the ''k''-th face r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed 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 theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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