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Pro-object
In mathematics, the ind-completion or ind-construction is the process of freely adding filtered colimits to a given category ''C''. The objects in this ind-completed category, denoted Ind(''C''), are known as direct systems, they are functors from a small filtered category ''I'' to ''C''. The dual concept is the pro-completion, Pro(''C''). Definitions Filtered categories Direct systems depend on the notion of ''filtered categories''. For example, the category N, whose objects are natural numbers, and with exactly one morphism from ''n'' to ''m'' whenever n \le m, is a filtered category. Direct systems A ''direct system'' or an ''ind-object'' in a category ''C'' is defined to be a functor :F : I \to C from a small filtered category ''I'' to ''C''. For example, if ''I'' is the category N mentioned above, this datum is equivalent to a sequence :X_0 \to X_1 \to \cdots of objects in ''C'' together with morphisms as displayed. The ind-completion Ind-objects in ''C'' form a cate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Finite Group
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups. The study of finite groups has been an integral part of group theory since it arose in the 19th century. One major area of study has been classification: the classification of finite simple groups (those with no nontrivial normal subgroup) was completed in 2004. History During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups. As a consequence, the complete classification of finite simple groups was achieved, meaning that all those simple groups from which all finite groups can be bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Deformation Theory
In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution ''P'' of a problem to slightly different solutions ''P''ε, where ε is a small number, or a vector of small quantities. The infinitesimal conditions are the result of applying the approach of differential calculus to solving a problem with constraints. The name is an analogy to non-rigid structures that deform slightly to accommodate external forces. Some characteristic phenomena are: the derivation of first-order equations by treating the ε quantities as having negligible squares; the possibility of ''isolated solutions'', in that varying a solution may not be possible, ''or'' does not bring anything new; and the question of whether the infinitesimal constraints actually 'integrate', so that their solution does provide small variations. In some form these considerations have a history of centuries in mathematics, but also in physics and engineering. For example, in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Schlessinger's Criterion
In algebra, Schlessinger's theorem is a theorem in deformation theory introduced by that gives conditions for a functor of artinian local rings to be pro-representable, refining an earlier theorem of Grothendieck. Definitions Λ is a complete Noetherian local ring with residue field ''k'', and ''C'' is the category of local Artinian Λ-algebras (meaning in particular that as modules over Λ they are finitely generated and Artinian) with residue field ''k''. A small extension in ''C'' is a morphism ''Y''→''Z'' in ''C'' that is surjective with kernel a 1-dimensional vector space over ''k''. A functor is called representable if it is of the form ''h''''X'' where ''h''''X''(''Y'')=hom(''X'',''Y'') for some ''X'', and is called pro-representable if it is of the form ''Y''→lim hom(''X''''i'',''Y'') for a filtered direct limit over ''i'' in some filtered ordered set. A morphism of functors ''F''→''G'' from ''C'' to sets is called smooth if whenever ''Y''→''Z'' is an epimorphi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Grothendieck's Galois Theory
In mathematics, Grothendieck's Galois theory is an abstract approach to the Galois theory of fields, developed around 1960 to provide a way to study the fundamental group of algebraic topology in the setting of algebraic geometry. It provides, in the classical setting of field theory, an alternative perspective to that of Emil Artin based on linear algebra, which became standard from about the 1930s. The approach of Alexander Grothendieck is concerned with the category-theoretic properties that characterise the categories of finite ''G''-sets for a fixed profinite group ''G''. For example, ''G'' might be the group denoted \hat (see profinite integer), which is the inverse limit of the cyclic additive groups \Z/n\Z — or equivalently the completion of the infinite cyclic group \Z for the topology of subgroups of finite index. A finite ''G''-set is then a finite set ''X'' on which ''G'' acts through a quotient finite cyclic group, so that it is specified by giving some permu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Shape Theory (mathematics)
Shape theory is a branch of topology that provides a more global view of the topological spaces than homotopy theory. The two coincide on compacta dominated homotopically by finite polyhedra. Shape theory associates with the Čech homology theory while homotopy theory associates with the singular homology theory. Background Shape theory was invented and published by D. E. Christie in 1944; it was reinvented, further developed and promoted by the Polish mathematician Karol Borsuk in 1968. Actually, the name ''shape theory'' was coined by Borsuk. Warsaw circle Borsuk lived and worked in Warsaw, hence the name of one of the fundamental examples of the area, the Warsaw circle. . Math 205B-2012 Lecture Notes, University of California Riverside. Retrieved ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Category Of Vector Spaces
In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ring of integers Z, it is the same thing as the category of abelian groups. The category of right modules is defined in a similar way. One can also define the category of bimodules over a ring ''R'' but that category is equivalent to the category of left (or right) modules over the enveloping algebra of ''R'' (or over the opposite of that). Note: Some authors use the term module category for the category of modules. This term can be ambiguous since it could also refer to a category with a monoidal-category action. Properties The categories of left and right modules are abelian categories. These categories have enough projectives and enough injectives. Mitchell's embedding theorem states every abelian category arises as a full subcate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Power Set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. The powerset of is variously denoted as , , , \mathbb(S), or . Any subset of is called a ''family of sets'' over . Example If is the set , then all the subsets of are * (also denoted \varnothing or \empty, the empty set or the null set) * * * * * * * and hence the power set of is . Properties If is a finite set with the cardinality (i.e., the number of all elements in the set is ), then the number of all the subsets of is . This fact as well as the reason of the notation denoting the power set are demonstrated in the below. : An indicator function or a characteristic function of a subset of a set with the cardinality is a function from to the two-element set , denoted as , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Stone Space
In topology and related areas of mathematics, a Stone space, also known as a profinite space or profinite set, is a compact Hausdorff totally disconnected space. Stone spaces are named after Marshall Harvey Stone who introduced and studied them in the 1930s in the course of his investigation of Boolean algebras, which culminated in his representation theorem for Boolean algebras. Equivalent conditions The following conditions on the topological space X are equivalent: * X is a Stone space; * X is homeomorphic to the projective limit (in the category of topological spaces) of an inverse system of finite discrete spaces; * X is compact and totally separated; * X is compact, T0, and zero-dimensional (in the sense of the small inductive dimension); * X is coherent and Hausdorff. Examples Important examples of Stone spaces include finite discrete spaces, the Cantor set and the space \Z_p of p-adic integers, where p is any prime number. Generalizing these examples, any p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Stone Duality
In mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the label Stone duality, since they form a natural generalization of Stone's representation theorem for Boolean algebras. These concepts are named in honor of Marshall Stone. Stone-type dualities also provide the foundation for pointless topology and are exploited in theoretical computer science for the study of formal semantics. This article gives pointers to special cases of Stone duality and explains a very general instance thereof in detail. Overview of Stone-type dualities Probably the most general duality that is classically referred to as "Stone duality" is the duality between the category Sob of sober spaces with continuous functions and the category SFrm of spatial frames with appropriate frame homomorphisms. The dual category of SFrm is the category of spatial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Spectral Space
In mathematics, a spectral space is a topological space that is homeomorphic to the spectrum of a commutative ring. It is sometimes also called a coherent space because of the connection to coherent topoi. Definition Let ''X'' be a topological space and let ''K''\circ(''X'') be the set of all compact open subsets of ''X''. Then ''X'' is said to be ''spectral'' if it satisfies all of the following conditions: *''X'' is compact and T0. * ''K''\circ(''X'') is a basis of open subsets of ''X''. * ''K''\circ(''X'') is closed under finite intersections. * ''X'' is sober, i.e., every nonempty irreducible closed subset of ''X'' has a (necessarily unique) generic point. Equivalent descriptions Let ''X'' be a topological space. Each of the following properties are equivalent to the property of ''X'' being spectral: #''X'' is homeomorphic to a projective limit of finite T0-spaces. #''X'' is homeomorphic to the spectrum of a bounded distributive lattice ''L''. In this case, ''L' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
