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Allegory (category Theory)
In the mathematical field of category theory, an allegory is a category that has some of the structure of the category Rel of sets and binary relations between them. Allegories can be used as an abstraction of categories of relations, and in this sense the theory of allegories is a generalization of relation algebra to relations between different sorts. Allegories are also useful in defining and investigating certain constructions in category theory, such as exact completions. In this article we adopt the convention that morphisms compose from right to left, so means "first do , then do ". Definition An allegory is a category in which * every morphism R\colon X\to Y is associated with an anti-involution, i.e. a morphism R^\circ\colon Y\to X with R^ = R and (RS)^\circ = S^\circ R^\circ\text and * every pair of morphisms R,S \colon X\to Y with common domain/codomain is associated with an intersection, i.e. a morphism R \cap S\colon X\to Y all such that * intersections are idem ... [...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 com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Span (category Theory)
In category theory, a span, roof or correspondence is a generalization of the notion of relation between two objects of a category. When the category has all pullbacks (and satisfies a small number of other conditions), spans can be considered as morphisms in a category of fractions. The notion of a span is due to Nobuo Yoneda (1954) and Jean Bénabou (1967). Formal definition A span is a diagram of type \Lambda = (-1 \leftarrow 0 \rightarrow +1), i.e., a diagram of the form Y \leftarrow X \rightarrow Z. That is, let Λ be the category (-1 ← 0 → +1). Then a span in a category ''C'' is a functor ''S'' : Λ → ''C''. This means that a span consists of three objects ''X'', ''Y'' and ''Z'' of ''C'' and morphisms ''f'' : ''X'' → ''Y'' and ''g'' : ''X'' → ''Z'': it is two maps with common ''domain''. The colimit of a span is a pushout. Examples * If ''R'' is a relation between sets ''X'' and ''Y'' (i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Terminal Object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): is terminal if for every object in there exists exactly one morphism . Initial objects are also called coterminal or universal, and terminal objects are also called final. If an object is both initial and terminal, it is called a zero object or null object. A pointed category is one with a zero object. A strict initial object is one for which every morphism into is an isomorphism. Examples * The empty set is the unique initial object in Set, the category of sets. Every one-element set ( singleton) is a terminal object in this category; there are no zero objects. Similarly, the empty space is the unique initial object in Top, the category of topological spaces and every one-point space is a terminal object in this category. * In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equalizer (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 any t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'Codomain. ''Encyclopedia of Mathematics''/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subcategory
In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, a subcategory of ''C'' is a category obtained from ''C'' by "removing" some of its objects and arrows. Formal definition Let ''C'' be a category. A subcategory ''S'' of ''C'' is given by *a subcollection of objects of ''C'', denoted ob(''S''), *a subcollection of morphisms of ''C'', denoted hom(''S''). such that *for every ''X'' in ob(''S''), the identity morphism id''X'' is in hom(''S''), *for every morphism ''f'' : ''X'' → ''Y'' in hom(''S''), both the source ''X'' and the target ''Y'' are in ob(''S''), *for every pair of morphisms ''f'' and ''g'' in hom(''S'') the composite ''f'' o ''g'' is in hom(''S'') whenever it is defined. These conditions ensure that ''S'' is a category in its own right: its collection of objects is ob(''S'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2-category
In category theory, a strict 2-category is a category with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. It can be formally defined as a category enriched over Cat (the category of categories and functors, with the monoidal structure given by product of categories). The concept of 2-category was first introduced by Charles Ehresmann in his work on enriched categories in 1965. The more general concept of bicategory (or ''weak'' 2-''category''), where composition of morphisms is associative only up to a 2-isomorphism, was introduced in 1968 by Jean Bénabou. Jean Bénabou, Introduction to bicategories, in Reports of the Midwest Category Seminar, Springer, Berlin, 1967, pp. 1--77. Definition A 2-category C consists of: * A class of 0-''cells'' (or ''objects'') , , .... * For all objects and , a category \mathbf(A,B). The objects f,g: A \to B of this category are called 1-''cells'' and its morphisms \alpha: f \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Functor
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e., constructions of objects having a certain universal property), such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology. By definition, an adjunction between categories \mathcal and \mathcal is a pair of functors (assumed to be covariant) :F: \mathcal \rightarrow \mathcal and G: \mathcal \rightarrow \mathcal and, for all objects X in \mathcal and Y in \mathcal a bijection between the respective morphism ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subobject
In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion is a generalization of concepts such as subsets from set theory, subgroups from group theory,Mac Lane, p. 126 and subspaces from topology. Since the detailed structure of objects is immaterial in category theory, the definition of subobject relies on a morphism that describes how one object sits inside another, rather than relying on the use of elements. The dual concept to a subobject is a . This generalizes concepts such as quotient sets, quotient groups, quotient spaces, quotient graphs, etc. Definitions In detail, let ''A'' be an object of some category. Given two monomorphisms :u: S \to A \ \text \ v: T\to A with codomain ''A'', we define an equivalence relation by u \equiv v if there exists an isomorphism \phi: S \to T with u = v \circ \phi. Equivalently, we write u \leq v if u factors through ''v''—that is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorization System
In mathematics, it can be shown that every function can be written as the composite of a surjective function followed by an injective function. Factorization systems are a generalization of this situation in category theory. Definition A factorization system (''E'', ''M'') for a category C consists of two classes of morphisms ''E'' and ''M'' of C such that: #''E'' and ''M'' both contain all isomorphisms of C and are closed under composition. #Every morphism ''f'' of C can be factored as f=m\circ e for some morphisms e\in E and m\in M. #The factorization is ''functorial'': if u and v are two morphisms such that vme=m'e'u for some morphisms e, e'\in E and m, m'\in M, then there exists a unique morphism w making the following diagram commute: ''Remark:'' (u,v) is a morphism from me to m'e' in the arrow category. Orthogonality Two morphisms e and m are said to be ''orthogonal'', denoted e\downarrow m, if for every pair of morphisms u and v such that ve=mu there is a unique mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pullback (category Theory)
In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms and with a common codomain. The pullback is often written : and comes equipped with two natural morphisms and . The pullback of two morphisms and need not exist, but if it does, it is essentially uniquely defined by the two morphisms. In many situations, may intuitively be thought of as consisting of pairs of elements with in , in , and . For the general definition, a universal property is used, which essentially expresses the fact that the pullback is the "most general" way to complete the two given morphisms to a commutative square. The dual concept of the pullback is the '' pushout''. Universal property Explicitly, a pullback of the morphisms and consists of an object and two morphisms and for which the diagram : commutes. Moreover, the pullback must be universal wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. The categorical dual of a monomorphism is an epimorphism, that is, a monomorphism in a category ''C'' is an epimorphism in the dual category ''C''op. Every section is a monomorphism, and every retraction is an epimorphism. Relation to invertibility Left-invertible morphisms are necessarily monic: if ''l'' is a left inverse for ''f'' (meanin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
