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Topological Topos
In mathematics, a generalized space is a generalization of a topological space. Impetuses for such a generalization comes at least in two forms: # A desire to apply concepts like cohomology for objects that are not traditionally viewed as spaces. For example, a topos was originally introduced for this reason. # A practical need to remedy the deficiencies that some naturally-occurring categories of spaces (e.g., ones in functional analysis) tend not to be abelian, a standard requirement to do homological algebra. Alexander Grothendieck's dictum says a topos is a generalized space; precisely, he and his followers write in exposé 4 of SGA I: However, William Lawvere argues in his 1975 paper that this dictum should be turned backward; namely, "a topos is the 'algebra of continuous (set-valued) functions' on a generalized space, not the generalized space itself." A generalized space should not be confused with a geometric object that can substitute the role of spaces. For example, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean Giraud (mathematician)
Jean Giraud (; 2 February 1936 – 27 or 28 March 2007) , Philippe Gillet, ''ENS Info'' 70, April 2007. was a French mathematician, a student of Alexander Grothendieck. His research focused on non-abelian cohomology and the theory of . In particular, he authored the book ''Cohomologie non-abélienne'' (Springer, 1971) and proved the theorem that bears his name, which gives a characterization of a Grothendieck topos. From 1969 to 1989, he was a professor at |
Completions In Category Theory
In category theory, a branch of mathematics, there are several ways (completions) to enlarge a given category in a way somehow analogous to a completion in topology. These are: (ignoring the set-theoretic matters for simplicity), *free cocompletion, free completion. These are obtained by freely adding colimits or limits. Explicitly, the free cocompletion of a category ''C'' is the Yoneda embedding of ''C'' into the category of presheaves on ''C''. The free completion of ''C'' is the free cocompletion of the opposite of ''C''. ** ind-completion. This is obtained by freely adding filtered colimits. *Cauchy completion of a category ''C'' is roughly the closure of ''C'' in some ambient category so that all functors preserve limits. For example, if a metric space is viewed as an enriched category (see generalized metric space), then the Cauchy completion of it coincides with the usual completion of the space. *Isbell completion (also called reflexive completion), introduced by Isbell in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Function
In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing discrete physical phenomena such as point charges. They are applied extensively, especially in physics and engineering. A common feature of some of the approaches is that they build on operator aspects of everyday, numerical functions. The early history is connected with some ideas on operational calculus, and more contemporary developments in certain directions are closely related to ideas of Mikio Sato, on what he calls algebraic analysis. Important influences on the subject have been the technical requirements of theories of partial differential equations, and group representation theory. Some early history In the mathematics of the nineteenth century, aspects of generalized function theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pyknotic Set
In mathematics, especially in topology, a pyknotic set is a sheaf of sets on the site of compact Hausdorff spaces (with some fixed Grothendieck universes). The notion was introduced by Barwick and Haine to provide a convenient setting for homological algebra. The term ''pyknotic'' comes from the Greek πυκνός, meaning dense, compact or thick. The notion can be compared to other approaches of introducing generalized spaces for the purpose of homological algebra such as Clausen and Scholze‘s condensed sets or Johnstone‘s topological topos In mathematics, a generalized space is a generalization of a topological space. Impetuses for such a generalization comes at least in two forms: # A desire to apply concepts like cohomology for objects that are not traditionally viewed as spaces. Fo .... Pyknotic sets form a coherent topos, while condensed sets do not. Comparing pyknotic sets with his approach with Clausen, Scholze writes: References * *Peter Scholze, Lectures on Co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Condensed Set
Condensed mathematics is a theory developed by and Peter Scholze which, according to some, aims to unify various mathematical subfields, including topology, complex geometry, and algebraic geometry. Idea The fundamental idea in the development of the theory is given by replacing topological spaces by ''condensed sets'', defined below. The category of condensed sets, as well as related categories such as that of condensed abelian groups, are much better behaved than the category of topological spaces. In particular, unlike the category of topological abelian groups, the category of condensed abelian groups is an abelian category, which allows for the use of tools from homological algebra in the study of those structures. The framework of condensed mathematics turns out to be general enough that, by considering various "spaces" with sheaves valued in condensed algebras, one is able to incorporate algebraic geometry, p-adic analytic geometry and complex analytic geometry. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Johnstone (mathematician)
Peter Tennant Johnstone (born 1948) is Professor of the Foundations of Mathematics at the University of Cambridge, and a fellow of St. John's College. He invented or developed a broad range of fundamental ideas in topos theory. His thesis, completed at the University of Cambridge in 1974, was entitled "Some Aspects of Internal Category Theory in an Elementary Topos". He is a great-great nephew of the Reverend George Gilfillan who was eulogised in William McGonagall William Topaz McGonagall (March 1825 – 29 September 1902) was a Scottish poet of Irish descent. He gained notoriety as an extremely bad poet who exhibited no recognition of, or concern for, his peers' opinions of his work. He wrote about 2 ...'s first poem. Books *. :— " r too hard to read, and not for the faint-hearted"An anonymous referee, as quoted by Johnstone in his ''Sketches of an elephant'', p. ix. *. *. * (v.3 in preparation) References External linksJohnstone's web page* * {{DEFAULTS ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gros Topos
{{disambiguation ...
Gros may refer to: People *Gros (surname) * Gross (surname), the German variant of Gros * Le Gros, the Norman variant of Gros Other uses * Gros (coinage), a type of 13th-century silver coinage of France * Gros (grape), another name for Elbling, a variety of white grape * Groș, a village of the city of Hunedoara, Transylvania, Romania * General Register Office for Scotland (GROS) See also * Gros Morne (other) * * Gross (other) * Grosz (other) Grosz may refer to: * Grosz, a coin valued as a hundredth of a Polish złoty * Kraków grosz, 14th-century coins of Kraków *Grosz (surname) See also * Gros (other) * Gross (other) Gross may refer to: Finance *Gross Cash ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locale (mathematics)
In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra that is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category CHey, the category Loc of locales, and its opposite, the category Frm of frames. Although these three categories contain the same objects, they differ in their morphisms, and thus get distinct names. Only the morphisms of CHey are homomorphisms of complete Heyting algebras. Locales and frames form the foundation of pointless topology, which, instead of building on point-set topology, recasts the ideas of general topology in categorical terms, as statements on frames and locales. Definition Consider a partially ordered set (''P'', ≤) that is a complete lattice. Then ''P'' is a complete Heyting algebra or frame if any of the following equivalent conditions hold: * ''P'' is a Heyting algebra, i.e. the operation (x\land\cdot) has a right adjoint (also called the lower adjo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory. From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic level, this has to do ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stack (mathematics)
In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory, and to construct fine moduli stacks when fine moduli spaces do not exist. Descent theory is concerned with generalisations of situations where isomorphic, compatible geometrical objects (such as vector bundles on topological spaces) can be "glued together" within a restriction of the topological basis. In a more general set-up the restrictions are replaced with pullbacks; fibred categories then make a good framework to discuss the possibility of such gluing. The intuitive meaning of a stack is that it is a fibred category such that "all possible gluings work". The specification of gluings requires a definition of coverings with regard to which the gluings can be considered. It turns out that the general language for describing these coverings is that of a Grothendieck topology. Thus a stack ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Lawvere
Francis William Lawvere (; born February 9, 1937) is a mathematician known for his work in category theory, topos theory and the philosophy of mathematics. Biography Lawvere studied continuum mechanics 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, following lectures by Alfred Tarski and Dana Scott. In his first teaching position at Reed College he was instructed to devise courses in calculus and abstract algebra from a foundational persp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
