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Differentiable Stack
A differentiable stack is the analogue in differential geometry of an algebraic stack in algebraic geometry. It can be described either as a stack over differentiable manifolds which admits an atlas, or as a Lie groupoid up to Morita equivalence. Differentiable stacks are particularly useful to handle spaces with singularities (i.e. orbifolds, leaf spaces, quotients), which appear naturally in differential geometry but are not differentiable manifolds. For instance, differentiable stacks have applications in foliation theory, Poisson geometry and twisted K-theory. Definition Definition 1 (via groupoid fibrations) Recall that a category fibred in groupoids (also called a groupoid fibration) consists of a category \mathcal together with a functor \pi: \mathcal \to \mathrm to the category of differentiable manifolds such that # \mathcal is a fibred category, i.e. for any object u of \mathcal and any arrow V \to U of \mathrm there is an arrow v \to u lying over V \to U; # for e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Descent (mathematics)
In mathematics, the idea of descent extends the intuitive idea of 'gluing' in topology. Since the topologists' glue is the use of equivalence relations on topological spaces, the theory starts with some ideas on identification. Descent of vector bundles The case of the construction of vector bundles from data on a disjoint union of topological spaces is a straightforward place to start. Suppose ''X'' is a topological space covered by open sets ''Xi''. Let ''Y'' be the disjoint union of the ''Xi'', so that there is a natural mapping :p: Y \rightarrow X. We think of ''Y'' as 'above' ''X'', with the ''Xi'' projection 'down' onto ''X''. With this language, ''descent'' implies a vector bundle on ''Y ''(so, a bundle given on each ''Xi''), and our concern is to 'glue' those bundles ''Vi'', to make a single bundle ''V'' on X. What we mean is that ''V'' should, when restricted to ''Xi'', give back ''Vi'', up to a bundle isomorphism. The data needed is then this: on each overlap :X_, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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71 may refer to: * 71 (number) * one of the years 71 BC, AD 71, 1971, 2071 * 71'' (film), 2014 British film set in Belfast in 1971 * '' 71: Into the Fire'', 2010 South Korean film See also * List of highways numbered A ''list'' is any set of items in a row. List or lists may also refer to: People * List (surname) Organizations * List College, an undergraduate division of the Jewish Theological Seminary of America * SC Germania List, German rugby union ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, ''morphism'' is a broadly similar idea: the mathematical objects involved need not be sets, and the relationships between them may be something other than maps, although the morphisms between the objects of a given category have to behave similarly to maps in that they have to admit an associative operation similar to function composition. A morphism in category theory is an abstraction of a homomorphism. The study of morphisms and of the structures (called "objects") over which they are defined is central to category theory. Much of the terminology of morphisms, as well as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grothendieck Topology
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site. Grothendieck topologies axiomatize the notion of an open cover. Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology. This was first done in algebraic geometry and algebraic number theory by Alexander Grothendieck to define the étale cohomology of a scheme. It has been used to define other cohomology theories since then, such as ℓ-adic cohomology, flat cohomology, and crystalline cohomology. While Grothendieck topologies are most often used to define cohomology theories, they have found other applications as well, such as to John Tate's theory of rigid analytic geometry. There is a natural way to associate a site to an ordinary top ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Site (mathematics)
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site. Grothendieck topologies axiomatize the notion of an open cover. Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology. This was first done in algebraic geometry and algebraic number theory by Alexander Grothendieck to define the étale cohomology of a scheme. It has been used to define other cohomology theories since then, such as ℓ-adic cohomology, flat cohomology, and crystalline cohomology. While Grothendieck topologies are most often used to define cohomology theories, they have found other applications as well, such as to John Tate's theory of rigid analytic geometry. There is a natural way to associate a site to an ordinary topol ... [...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] |
2 Category
In category theory, a strict 2-category is a category with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. It can be formally defined as a category enriched over Cat (the category of categories and functors, with the monoidal structure given by product of categories). The concept of 2-category was first introduced by Charles Ehresmann in his work on enriched categories in 1965. The more general concept of bicategory (or ''weak'' 2-''category''), where composition of morphisms is associative only up to a 2-isomorphism, was introduced in 1968 by Jean Bénabou.Jean Bénabou, Introduction to bicategories, in Reports of the Midwest Category Seminar, Springer, Berlin, 1967, pp. 1--77. Definition A 2-category C consists of: * A class of 0-''cells'' (or ''objects'') , , .... * For all objects and , a category \mathbf(A,B). The objects f,g: A \to B of this category are called 1-''cells'' and its morphisms \alpha: f \Righ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2-functor
In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ..., a 2-functor is a morphism between 2-categories. They may be defined formally using enrichment by saying that a 2-category is exactly a ''Cat''-enriched category and a 2-functor is a ''Cat''-functor. Explicitly, if ''C'' and ''D'' are 2-categories then a 2-functor F\colon C\to D consists of * a function F\colon \text C\to \text D, and * for each pair of objects c,c'\in C a functor F_\colon \text_(c,c')\to\text_D(Fc,Fc') such that each F_ strictly preserves identity objects and they commute with horizontal composition in ''C'' and ''D''. See for more details and for lax versions. References Functors Higher category theory {{categorytheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Presheaf (category Theory)
In category theory, a branch of mathematics, a presheaf on a category C is a functor F\colon C^\mathrm\to\mathbf. If C is the poset of open sets in a topological space, interpreted as a category, then one recovers the usual notion of presheaf on a topological space. A morphism of presheaves is defined to be a natural transformation of functors. This makes the collection of all presheaves on C into a category, and is an example of a functor category. It is often written as \widehat = \mathbf^. A functor into \widehat is sometimes called a profunctor. A presheaf that is naturally isomorphic to the contravariant hom-functor Hom(–, ''A'') for some object ''A'' of C is called a representable presheaf. Some authors refer to a functor F\colon C^\mathrm\to\mathbf as a \mathbf-valued presheaf. Examples * A simplicial set is a Set-valued presheaf on the simplex category C=\Delta. Properties * When C is a small category, the functor category \widehat=\mathbf^ is cartesian closed. * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prestack
In algebraic geometry, a prestack ''F'' over a category ''C'' equipped with some Grothendieck topology is a category together with a functor ''p'': ''F'' → ''C'' satisfying a certain lifting condition and such that (when the fibers are groupoids) locally isomorphic objects are isomorphic. A stack is a prestack with effective descents, meaning local objects may be patched together to become a global object. Prestacks that appear in nature are typically stacks but some naively constructed prestacks (e.g., groupoid scheme or the prestack of projectivized vector bundles) may not be stacks. Prestacks may be studied on their own or passed to stacks. Since a stack is a prestack, all the results on prestacks are valid for stacks as well. Throughout the article, we work with a fixed base category ''C''; for example, ''C'' can be the category of all schemes over some fixed scheme equipped with some Grothendieck topology. Informal definition Let ''F'' be a category and suppose it is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surjective Function
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of its domain. It is not required that be unique; the function may map one or more elements of to the same element of . The term ''surjective'' and the related terms ''injective'' and ''bijective'' were introduced by Nicolas Bourbaki, a group of mainly French 20th-century mathematicians who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word '' sur'' means ''over'' or ''above'', and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain. Any function induces a surjection by restricting its codomain to the image of its domain. Every surjective function has a right inverse assuming the axiom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
