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Nodal Decomposition
In category theory, an abstract mathematical discipline, a nodal decomposition of a morphism \varphi:X\to Y is a representation of \varphi as a product \varphi=\sigma\circ\beta\circ\pi, where \pi is a Epimorphism#Related concepts, strong epimorphism, \beta a bimorphism, and \sigma a Monomorphism#Related concepts, strong monomorphism.A monomorphism \mu:C\to D is said to be strong, if for any epimorphism \varepsilon:A\to B and for any morphisms \alpha:A\to C and \beta:B\to D such that \beta\circ\varepsilon=\mu\circ\alpha there exists a morphism \delta:B\to C, such that \delta\circ\varepsilon=\alpha and \mu\circ\delta=\beta Uniqueness and notations If it exists, the nodal decomposition is unique up to an isomorphism in the following sense: for any two nodal decompositions \varphi=\sigma\circ\beta\circ\pi and \varphi=\sigma'\circ\beta'\circ\pi' there exist Morphism#Isomorphisms, isomorphisms \eta and \theta such that : \pi'=\eta\circ\pi, : \beta=\theta\circ\beta'\circ\eta, : \sigm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Additive Category
In mathematics, specifically in category theory, an additive category is a preadditive category C admitting all finitary biproducts. Definition There are two equivalent definitions of an additive category: One as a category equipped with additional structure, and another as a category equipped with ''no extra structure'' but whose objects and morphisms satisfy certain equations. Via preadditive categories A category C is preadditive if all its hom-sets are abelian groups and composition of morphisms is bilinear; in other words, C is enriched over the monoidal category of abelian groups. In a preadditive category, every finitary product is necessarily a coproduct, and hence a biproduct, and conversely every finitary coproduct is necessarily a product (this is a consequence of the definition, not a part of it). The empty product, is a final object and the empty product in the case of an empty diagram, an initial object. Both being limits, they are not finite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Stereotype Space
In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is a homeomorphism (or equivalently, a TVS isomorphism). A normed space is reflexive if and only if this canonical evaluation map is surjective, in which case this (always linear) evaluation map is an isometric isomorphism and the normed space is a Banach space. Those spaces for which the canonical evaluation map is surjective are called semi-reflexive spaces. In 1951, R. C. James discovered a Banach space, now known as James' space, that is reflexive (meaning that the canonical evaluation map is not an isomorphism) but is nevertheless isometrically isomorphic to its bidual (any such isometric isomorphism is necessarily the canonical evaluation map). So importantly, for a Banach space to be reflexive, it is not enough for it to be isometricall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Strong Epimorphism
In category theory, an epimorphism is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f \implies g_1 = g_2. Epimorphisms are categorical analogues of onto or surjective functions (and in the category of sets the concept corresponds exactly to the surjective functions), but they may not exactly coincide in all contexts; for example, the inclusion \mathbb\to\mathbb is a ring epimorphism. The dual of an epimorphism is a monomorphism (i.e. an epimorphism in a category ''C'' is a monomorphism in the dual category ''C''op). Many authors in abstract algebra and universal algebra define an epimorphism simply as an ''onto'' or surjective homomorphism. Every epimorphism in this algebraic sense is an epimorphism in the sense of category theory, but the converse is not true in all categories. In this article, the term "epimorphism" will be used in the sense of category theory given above ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Strong Monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism. That is, an arrow such that for all objects and all morphisms , : f \circ g_1 = f \circ g_2 \implies g_1 = g_2. Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below. In the setting of posets intersections are idempotent: the intersection of anything with itself is itself. Monomorphisms generalize this property to arbitrary categories. A morphism is a monomorphism if it is idempotent with respect to pullbacks. The categorical dual of a monomorphism is an epimorphism, that is, a monomorphism in a category ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Inverse Limit
In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits can be defined in any category although their existence depends on the category that is considered. They are a special case of the concept of limit in category theory. By working in the dual category, that is by reversing the arrows, an inverse limit becomes a direct limit or ''inductive limit'', and a ''limit'' becomes a colimit. Formal definition Algebraic objects We start with the definition of an inverse system (or projective system) of groups and homomorphisms. Let (I, \leq) be a directed poset (not all authors require ''I'' to be directed). Let (''A''''i'')''i''∈''I'' be a family of groups and suppose we have a family of homomorphisms f_: A_j \to A_i for all i \leq j (note the order) with the following properties: # f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Direct Limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any category. The way they are put together is specified by a system of homomorphisms (group homomorphism, ring homomorphism, or in general morphisms in the category) between those smaller objects. The direct limit of the objects A_i, where i ranges over some directed set I, is denoted by \varinjlim A_i . This notation suppresses the system of homomorphisms; however, the limit depends on the system of homomorphisms. Direct limits are a special case of the concept of colimit in category theory. Direct limits are dual to inverse limits, which are a special case of limits in category theory. Formal definition We will first give the definition for algebraic structures like groups and modules, and then the general definition, which can be used ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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, . 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 Groth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Refinement (category Theory)
In category theory and related fields of mathematics, a refinement is a construction that generalizes the operations of "interior enrichment", like bornologification or saturation of a locally convex space. A dual construction is called Envelope (category theory), envelope. Definition Suppose K is a category, X an object in K, and \Gamma and \Phi two classes of morphisms in K. The definition of a refinement of X in the class \Gamma by means of the class \Phi consists of two steps. * A morphism \sigma:X'\to X in K is called an ''enrichment of the object X in the class of morphisms \Gamma by means of the class of morphisms \Phi'', if \sigma\in\Gamma, and for any morphism \varphi:B\to X from the class \Phi there exists a unique morphism \varphi':B\to X' in K such that \varphi=\sigma\circ\varphi'. * An enrichment \rho:E\to X of the object X in the class of morphisms \Gamma by means of the class of morphisms \Phi is called a ''refinement of X in \Gamma by means of \Phi'', if for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Envelope (category Theory)
In category theory and related fields of mathematics, an envelope is a construction that generalizes the operations of "exterior completion", like completion of a locally convex space, or Stone–Čech compactification of a topological space. A dual construction is called Refinement (category theory), refinement. Definition Suppose K is a category, X an object in K, and \Omega and \Phi two classes of morphisms in K. The definition of an envelope of X in the class \Omega with respect to the class \Phi consists of two steps. * A morphism \sigma:X\to X' in K is called an ''extension of the object X in the class of morphisms \Omega with respect to the class of morphisms \Phi'', if \sigma\in\Omega, and for any morphism \varphi:X\to B from the class \Phi there exists a unique morphism \varphi':X'\to B in K such that \varphi=\varphi'\circ\sigma. * An extension \rho:X\to E of the object X in the class of morphisms \Omega with respect to the class of morphisms \Phi is called an ''enve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Pre-abelian Category
In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels. Spelled out in more detail, this means that a category C is pre-abelian if: # C is preadditive, that is enriched over the monoidal category of abelian groups (equivalently, all hom-sets in C are abelian groups and composition of morphisms is bilinear); # C has all finite products (equivalently, all finite coproducts); note that because C is also preadditive, finite products are the same as finite coproducts, making them biproducts; # given any morphism ''f'': ''A'' → ''B'' in C, the equaliser of ''f'' and the zero morphism from ''A'' to ''B'' exists (this is by definition the kernel of ''f''), as does the coequaliser (this is by definition the cokernel of ''f''). Note that the zero morphism in item 3 can be identified as the identity element of the hom-set Hom(''A'',''B''), which is an abelian group by item 1; or as the unique mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
