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Cochain
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is contained in the kernel of the next. Associated to a chain complex is its homology, which is (loosely speaking) a measure of the failure of a chain complex to be exact. A cochain complex is similar to a chain complex, except that its homomorphisms are in the opposite direction. The homology of a cochain complex is called its cohomology. In algebraic topology, the singular chain complex of a topological space X is constructed using continuous maps from a simplex to X, and the homomorphisms of the chain complex capture how these maps restrict to the boundary of the simplex. The homology of this chain complex is called the singular homology of X, and is a commonly used invariant of a topological space. Chain complexes are studied in homological algebra, but ar ... [...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 function (mathematics), functions on the group of chain (algebraic topology), chains in homology theory. From its start 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 abstract algebra, algebra. The terminology tends to hide the fact that cohomology, a Covariance and contravariance of functors, c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chain Map
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is contained in the kernel of the next. Associated to a chain complex is its homology, which is (loosely speaking) a measure of the failure of a chain complex to be exact. A cochain complex is similar to a chain complex, except that its homomorphisms are in the opposite direction. The homology of a cochain complex is called its cohomology. In algebraic topology, the singular chain complex of a topological space X is constructed using continuous maps from a simplex to X, and the homomorphisms of the chain complex capture how these maps restrict to the boundary of the simplex. The homology of this chain complex is called the singular homology of X, and is a commonly used invariant of a topological space. Chain complexes are studied in homological algebra, but ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Homology
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension n, the n-dimensional holes of a space. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions (see also the related theory simplicial homology). In brief, singular homology is constructed by taking maps of the simplex, standard -simplex to a topological space, and composing them into Free abelian group#Integer functions and formal sums, formal sums, called singular chains. The boundary operation – mapping each n-dimensional simplex to its (n-1)-dimensional boundary operator, boundary – induces the singular chain complex. The singular homology is then ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homology (mathematics)
In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages. The most direct usage of the term is to take the ''homology of a chain complex'', resulting in a sequence of Abelian group, abelian groups called ''homology groups.'' This operation, in turn, allows one to associate various named ''homologies'' or ''homology theories'' to various other types of mathematical objects. Lastly, since there are many homology theories for Topological space, topological spaces that produce the same answer, one also often speaks of the ''homology of a topological space''. (This latter notion of homology admits more intuitive descriptions for 1- or 2-dimensional topological spaces, and is sometimes referenced in popular mathematics.) There is also a related notion of the cohomology of a Cochain complexes, cochain complex, giving rise to various cohomology theories, in addition to the notion of the cohomology of a topological space. Ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Abelian Categories
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of abelian groups, . Abelian categories are very ''stable'' categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory. Mac Lane says Alexander Grothendieck defined the abelian category, but there is a reference that says Eilenberg's disciple, Buchsbaum, proposed the concept in his PhD thesis, and Grothendie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual (category Theory)
In category theory, a branch of mathematics, duality is a correspondence between the properties of a category ''C'' and the dual properties of the opposite category ''C''op. Given a statement regarding the category ''C'', by interchanging the source and target of each morphism as well as interchanging the order of composing two morphisms, a corresponding dual statement is obtained regarding the opposite category ''C''op. (''C''op is composed by reversing every morphism of ''C''.) Duality, as such, is the assertion that truth is invariant under this operation on statements. In other words, if a statement ''S'' is true about ''C'', then its dual statement is true about ''C''op. Also, if a statement is false about ''C'', then its dual has to be false about ''C''op. (Compactly saying, ''S'' for ''C'' is true if and only if its dual for ''C''op is true.) Given a concrete category ''C'', it is often the case that the opposite category ''C''op per se is abstract. ''C''op need not b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prefix
A prefix is an affix which is placed before the stem of a word. Particularly in the study of languages, a prefix is also called a preformative, because it alters the form of the word to which it is affixed. Prefixes, like other affixes, can be either inflectional, creating a new form of a word with the same basic meaning and same lexical category, or derivational, creating a new word with a new semantic meaning and sometimes also a different lexical category. Prefixes, like all affixes, are usually bound morphemes. English has no inflectional prefixes, using only suffixes for that purpose. Adding a prefix to the beginning of an English word changes it to a different word. For example, when the prefix ''un-'' is added to the word ''happy'', it creates the word ''unhappy''. The word ''prefix'' is itself made up of the stem ''fix'' (meaning "attach", in this case), and the prefix ''pre-'' (meaning "before"), both of which are derived from Latin roots. English language ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost All
In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if X is a set (mathematics), set, "almost all elements of X" means "all elements of X but those in a negligible set, negligible subset of X". The meaning of "negligible" depends on the mathematical context; for instance, it can mean finite set, finite, countable set, countable, or null set, null. In contrast, "almost no" means "a negligible quantity"; that is, "almost no elements of X" means "a negligible quantity of elements of X". Meanings in different areas of mathematics Prevalent meaning Throughout mathematics, "almost all" is sometimes used to mean "all (elements of an infinite set) except for finite set, finitely many". This use occurs in philosophy as well. Similarly, "almost all" can mean "all (elements of an uncountable set) except for countable set, countably many". Examples: * Almost all positive integers are greater than 1012. * Almost all prime numbers are odd (2 is the only ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Homology
In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of a given dimension in the complex. This generalizes the number of connected component (topology), connected components (the case of dimension 0). Simplicial homology arose as a way to study topological spaces whose building blocks are ''n''-simplices, the ''n''-dimensional analogs of triangles. This includes a point (0-simplex), a line segment (1-simplex), a triangle (2-simplex) and a tetrahedron (3-simplex). By definition, such a space is homeomorphic to a simplicial complex (more precisely, the abstract simplicial complex, geometric realization of an abstract simplicial complex). Such a homeomorphism is referred to as a ''Triangulation (topology), triangulation'' of the given space. Many topological spaces of interest can be triangulated, including every smooth manifold (Cairns and J.H.C. Whitehead, Whitehead). Simplicial homology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as classical antiquity, antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Nikolai Lobachevsky, Lobachevsky. The simplest examples of smooth spaces are the Differential geometry of curves, plane and space curves and Differential geometry of surfaces, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
