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Groupoid Object
In category theory, a branch of mathematics, a groupoid object is both a generalization of a groupoid which is built on richer structures than sets, and a generalization of a group objects when the multiplication is only partially defined. Definition A groupoid object in a category C admitting finite fiber products consists of a pair of objects R, U together with five morphisms :s, t: R \to U, \ e: U \to R, \ m: R \times_ R \to R, \ i: R \to R satisfying the following groupoid axioms # s \circ e = t \circ e = 1_U, \, s \circ m = s \circ p_1, t \circ m = t \circ p_2 where the p_i: R \times_ R \to R are the two projections, # (associativity) m \circ (1_R \times m) = m \circ (m \times 1_R), # (unit) m \circ (e \circ s, 1_R) = m \circ (1_R, e \circ t) = 1_R, # (inverse) i \circ i = 1_R, s \circ i = t, \, t \circ i = s, m \circ (1_R, i) = e \circ s, \, m \circ (i, 1_R) = e \circ t. Examples Group objects A group object is a special case of a groupoid object, where R = U and s = ... [...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] |
Scheme (mathematics)
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations and define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise '' Éléments de géométrie algébrique'' (EGA); one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem. Schemes elaborate the fundamental idea that an a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deligne–Mumford Stack
In algebraic geometry, a Deligne–Mumford stack is a stack ''F'' such that Pierre Deligne and David Mumford introduced this notion in 1969 when they proved that moduli spaces of stable curves of fixed arithmetic genus are proper smooth Deligne–Mumford stacks. If the "étale" is weakened to " smooth", then such a stack is called an algebraic stack (also called an Artin stack, after Michael Artin). An algebraic space is Deligne–Mumford. A key fact about a Deligne–Mumford stack ''F'' is that any ''X'' in F(B), where ''B'' is quasi-compact, has only finitely many automorphisms. A Deligne–Mumford stack admits a presentation by a groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: * '' Group'' with a partial fu ...; see groupoid scheme. Examples Affine Stacks Deligne–Mumford stacks are typica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Fibered In Groupoids
Stack may refer to: Places * Stack Island, an island game reserve in Bass Strait, south-eastern Australia, in Tasmania’s Hunter Island Group * Blue Stack Mountains, in Co. Donegal, Ireland People * Stack (surname) (including a list of people with the name) * Parnell "Stacks" Edwards, a key associate in the Lufthansa heist * Robert Stack Pierce (1933–2016), an American actor and baseball player * Robert Stack (1919 – 2003), and American actor and television show host * Brian "Stack" Stevens (1941–2017), a Cornish rugby player Arts, entertainment, and media * ''Stack magazine'', a bimonthly publication about high school sports * ''Stacks'' (album), a 2005 album by Bernie Marsden * Stacks, trailer parks that were made vertical, in the film ''Ready Player One'' Computing * Stack (abstract data type), abstract data type and data structure based on the principle of last in first out * Stack (Haskell), a tool to build Haskell projects and manage their dependencies * Stack ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torsor Under A Groupoid
In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non-empty set ''X'' on which ''G'' acts freely and transitively (meaning that, for any ''x'', ''y'' in ''X'', there exists a unique ''g'' in ''G'' such that , where · denotes the (right) action of ''G'' on ''X''). An analogous definition holds in other categories, where, for example, *''G'' is a topological group, ''X'' is a topological space and the action is continuous, *''G'' is a Lie group, ''X'' is a smooth manifold and the action is smooth, *''G'' is an algebraic group, ''X'' is an algebraic variety and the action is regular. Definition If ''G'' is nonabelian then one must distinguish between left and right torsors according to whether the action is on the left or right. In this article, we will use right actions. To state the defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atlas (stack)
An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a continent or region of Earth. Advances in astronomy have also resulted in atlases of the celestial sphere or of other planets. Atlases have traditionally been bound into book form, but today, many atlases are in multimedia formats. In addition to presenting geographical features and political boundaries, many atlases often feature geopolitical, social, religious, and economic statistics. They also have information about the map and places in it. Etymology The use of the word "atlas" in a geographical context dates from 1595 when the German-Flemish geographer Gerardus Mercator published ("Atlas or cosmographical meditations upon the creation of the universe and the universe as created"). This title provides Mercator's definition of the word as a description of the creation and form of the whole universe, not simply as a collection of maps. The volume that was published posthumously one year aft ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stackification
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] |
Stack (mathematics)
In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf (mathematics), sheaf that takes values in category (mathematics), 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 isomorphism, 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 Pullback (category theory), pullbacks; fibred category, 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 ... [...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] |
Contravariant Functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied. The words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used ''functor'' in a linguistic context; see function word. Definition Let ''C'' and ''D'' be categories. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each object X in ''C'' to an object F(X) in ''D'', * associates each morphism f \colon X \to Y in ''C'' to a morphism F(f) \colon F(X) \to F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coequalizer
In category theory, a coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation to objects in an arbitrary category. It is the categorical construction dual to the equalizer. Definition A coequalizer is the colimit of a diagram consisting of two objects ''X'' and ''Y'' and two parallel morphisms . More explicitly, a coequalizer of the parallel morphisms ''f'' and ''g'' can be defined as an object ''Q'' together with a morphism such that . Moreover, the pair must be universal in the sense that given any other such pair (''Q''′, ''q''′) there exists a unique morphism such that . This information can be captured by the following commutative diagram: As with all universal constructions, a coequalizer, if it exists, is unique up to a unique isomorphism (this is why, by abuse of language, one sometimes speaks of "the" coequalizer of two parallel arrows). It can be shown that a coequalizing arrow ''q'' is an epimorphism in any ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equaliser (mathematics)
In mathematics, an equaliser is a set of arguments where two or more functions have equal values. An equaliser is the solution set of an equation. In certain contexts, a difference kernel is the equaliser of exactly two functions. Definitions Let ''X'' and ''Y'' be sets. Let ''f'' and ''g'' be functions, both from ''X'' to ''Y''. Then the ''equaliser'' of ''f'' and ''g'' is the set of elements ''x'' of ''X'' such that ''f''(''x'') equals ''g''(''x'') in ''Y''. Symbolically: : \operatorname(f, g) := \. The equaliser may be denoted Eq(''f'', ''g'') or a variation on that theme (such as with lowercase letters "eq"). In informal contexts, the notation is common. The definition above used two functions ''f'' and ''g'', but there is no need to restrict to only two functions, or even to only finitely many functions. In general, if F is a set of functions from ''X'' to ''Y'', then the ''equaliser'' of the members of F is the set of elements ''x'' of ''X'' such that, given an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
