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Pseudo-abelian Category
In mathematics, specifically in category theory, a pseudo-abelian category is a category (mathematics), category that is preadditive category, preadditive and is such that every idempotent has a kernel (category theory), kernel. Recall that an idempotent morphism p is an endomorphism of an object with the property that p\circ p = p. Elementary considerations show that every idempotent then has a cokernel.Lars Brünjes, Forms of Fermat equations and their zeta functions, Appendix A The pseudo-abelian condition is stronger than preadditivity, but it is weaker than the requirement that every morphism have a kernel and cokernel, as is true for abelian categories. Synonyms in the literature for pseudo-abelian include pseudoabelian and Karoubian. Examples Any abelian category, in particular the category Ab of abelian groups, is pseudo-abelian. Indeed, in an abelian category, ''every'' morphism has a kernel. The category of associative rng (algebra), rngs (not ring (mathematics), rings ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. 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 spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism compos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. '' Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Preadditive Category
In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab. That is, an Ab-category C is a category such that every hom-set Hom(''A'',''B'') in C has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation. In formulas: f\circ (g + h) = (f\circ g) + (f\circ h) and (f + g)\circ h = (f\circ h) + (g\circ h), where + is the group operation. Some authors have used the term ''additive category'' for preadditive categories, but here we follow the current trend of reserving this term for certain special preadditive categories (see below). Examples The most obvious example of a preadditive category is the category Ab itself. More precisely, Ab is a closed monoidal category. Note that commutativity is crucial here; it ensures that the sum of two group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Idempotent
Idempotence (, ) is the property of certain operation (mathematics), operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in abstract algebra (in particular, in the theory of projector (linear algebra), projectors and closure operators) and functional programming (in which it is connected to the property of referential transparency). The term was introduced by American mathematician Benjamin Peirce in 1870 in the context of elements of algebras that remain invariant when raised to a positive integer power, and literally means "(the quality of having) the same power", from + ''wikt:potence, potence'' (same + power). Definition An element x of a set S equipped with a binary operator \cdot is said to be ''idempotent'' under \cdot if : . The ''binary operation'' \cdot is said to be ''idempotent'' if : . Examples * In the monoid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel (category Theory)
In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms, the kernels of module homomorphisms and certain other kernels from algebra. Intuitively, the kernel of the morphism ''f'' : ''X'' → ''Y'' is the "most general" morphism ''k'' : ''K'' → ''X'' that yields zero when composed with (followed by) ''f''. Note that kernel pairs and difference kernels (also known as binary equalisers) sometimes go by the name "kernel"; while related, these aren't quite the same thing and are not discussed in this article. Definition Let C be a category. In order to define a kernel in the general category-theoretical sense, C needs to have zero morphisms. In that case, if ''f'' : ''X'' → ''Y'' is an arbitrary morphism in C, then a kernel of ''f'' is an equaliser of ''f'' and the zero morphism from ''X'' to ''Y''. In symbols: :ker(''f'') = eq(''f'', 0''XY'') To be more explicit, the following universal pro ... [...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, Ab. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. 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. Abelian categories are named a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Category
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, Ab. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. 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. Abelian categories are na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Groups
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. Definition An abelian group is a set A, together with an operation \cdot that combines any two elements a and b of A to form another element of A, denoted a \cdot b. The symbo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rng (algebra)
In mathematics, and more specifically in abstract algebra, a rng (or non-unital ring or pseudo-ring) is an algebraic structure satisfying the same properties as a ring, but without assuming the existence of a multiplicative identity. The term ''rng'' (IPA: ) is meant to suggest that it is a ring without ''i'', that is, without the requirement for an identity element. There is no consensus in the community as to whether the existence of a multiplicative identity must be one of the ring axioms (see ). The term ''rng'' was coined to alleviate this ambiguity when people want to refer explicitly to a ring without the axiom of multiplicative identity. A number of algebras of functions considered in analysis are not unital, for instance the algebra of functions decreasing to zero at infinity, especially those with compact support on some (non-compact) space. Definition Formally, a rng is a set ''R'' with two binary operations called ''addition'' and ''multiplication'' such that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (mathematics)
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ''ring'' is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. Formally, a ''ring'' is an abelian group whose operation is called ''addition'', with a second binary operation called ''multiplication'' that is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors use the term " " with a missing i to refer to the more general structure that omits this last requirement; see .) Whether a ring is commutative (that is, whether the order in which two elements are multiplied might change the result) has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chow Motives
{{disambiguation ...
Chow may refer to: * Selected set of nutrients fed to animals subjected to laboratory testing * Chow Chow, a dog breed * A slang term for food in general (such as in the terms "chow down" or "chow hall") * Chow test, a statistical test for detecting differences between trends in time series * Chow (unit), an obsolete unit of mass in the pearl trade in Mumbai * Chow (website), a popular online food discussion site * Chow, an alternate name for the star Beta Serpentis * Mr. Chow, an upscale Chinese restaurant chain * Chow (surname), an English surname, as well as a Latin-alphabet spelling of various Chinese surnames See also * Ciao * Chew (other) * Chao (other) Chao may refer to: People * Chao (surname), various Chinese surnames (including 晁 and 巢, as well as non-Pinyin spellings) * Zhou (surname) (周), may also be spelled Chao * Zhao (surname) (趙/赵), may also be spelled Chao in Taiwan and Hon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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