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Extranatural Transformation
(dually co-wedges and co-ends), by setting F (dually G) constant. Extranatural transformations can be defined in terms of dinatural transformations, of which they are a special case. See also * Dinatural transformation External links * {{nlab, id=extranatural+transformation References Higher category theory ... [...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] |
Samuel Eilenberg
Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra. Early life and education He was born in Warsaw, Kingdom of Poland to a Jewish family. He spent much of his career as a professor at Columbia University. He earned his Ph.D. from University of Warsaw in 1936, with thesis ''On the Topological Applications of Maps onto a Circle''; his thesis advisors were Kazimierz Kuratowski and Karol Borsuk. He died in New York City in January 1998. Career Eilenberg's main body of work was in algebraic topology. He worked on the axiomatic treatment of homology theory with Norman Steenrod (and the Eilenberg–Steenrod axioms are named for the pair), and on homological algebra with Saunders Mac Lane. In the process, Eilenberg and Mac Lane created category theory. Eilenberg was a member of Bourbaki and, with Henri Cartan, wrote the 1956 book ''Homological Algebra''. Later ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Max Kelly
Gregory Maxwell "Max" Kelly (5 June 1930 – 26 January 2007) was an Australian mathematician who worked on category theory. Biography Kelly was born in Bondi, New South Wales, Australia, on 5 June 1930. He obtained his PhD at Cambridge University in homological algebra in 1957, publishing his first paper in that area in 1959, ''Single-space axioms for homology theory''. He taught in the Pure Mathematics department at the University of Sydney from 1957 to 1966, rising from lecturer to reader. During 1963–1965 he was a visiting fellow at Tulane University and the University of Illinois, where with Samuel Eilenberg he formalized and developed the notion of an enriched category based on intuitions then in the air about making the homsets of a category just as abstract as the objects themselves. He subsequently developed the notion in considerably more detail in his 1982 monograph ''Basic Concepts of Enriched Category Theory''. Let \cal V be a monoidal category, and denote ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Informally, the notion of a natural transformation states that a particular map between functors can be done consistently over an entire category. Indeed, this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications. Definition If F and G are functors between the categories C and D , then a natural transformation \eta from F to G is a family of morphisms that satisfies two requirements. # The natural transformation must associate, to every object X in C, a morphism \eta_X : F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous function, continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied. The words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used ''functor'' in a Linguistics, linguistic context; see function word. Definition Let ''C'' and ''D'' be category (mathematics), categories. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each object X in ''C'' to an object F(X) in ''D' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Diagram
350px, The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result. It is said that commutative diagrams play the role in category theory that equations play in algebra. Description A commutative diagram often consists of three parts: * objects (also known as ''vertices'') * morphisms (also known as ''arrows'' or ''edges'') * paths or composites Arrow symbols In algebra texts, the type of morphism can be denoted with different arrow usages: * A monomorphism may be labeled with a \hookrightarrow or a \rightarrowtail. * An epimorphism may be labeled with a \twoheadrightarrow. * An isomorphism may be labeled with a \overset. * The dashed arrow typically represents the claim that the indicated morphism exists (whenever the rest of the diagram holds); the arrow may be optionally labeled as \exist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
End (category Theory)
In category theory, an end of a functor S:\mathbf^\times\mathbf\to \mathbf is a universal extranatural transformation from an object ''e'' of X to ''S''. More explicitly, this is a pair (e,\omega), where ''e'' is an object of X and \omega:e\ddot\to S is an extranatural transformation such that for every extranatural transformation \beta : x\ddot\to S there exists a unique morphism h:x\to e of X with \beta_a=\omega_a\circ h for every object ''a'' of C. By abuse of language the object ''e'' is often called the ''end'' of the functor ''S'' (forgetting \omega) and is written :e=\int_c^ S(c,c)\text\int_\mathbf^ S. Characterization as limit: If X is complete and C is small, the end can be described as the equalizer in the diagram :\int_c S(c, c) \to \prod_ S(c, c) \rightrightarrows \prod_ S(c, c'), where the first morphism being equalized is induced by S(c, c) \to S(c, c') and the second is induced by S(c', c') \to S(c, c'). Coend The definition of the coend of a functor S:\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dinatural Transformation
In category theory, a branch of mathematics, a dinatural transformation \alpha between two functors :S,T : C^\times C\to D, written :\alpha : S\ddot\to T, is a function that to every object c of C associates an arrow :\alpha_c : S(c,c)\to T(c,c) of D and satisfies the following coherence property: for every morphism f:c\to c' of C the diagram commutes. The composition of two dinatural transformations need not be dinatural. See also * Extranatural transformation *Natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ... References External links dinatural transformationat the ''n''-Category Lab. Functors {{categorytheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dinatural Transformation
In category theory, a branch of mathematics, a dinatural transformation \alpha between two functors :S,T : C^\times C\to D, written :\alpha : S\ddot\to T, is a function that to every object c of C associates an arrow :\alpha_c : S(c,c)\to T(c,c) of D and satisfies the following coherence property: for every morphism f:c\to c' of C the diagram commutes. The composition of two dinatural transformations need not be dinatural. See also * Extranatural transformation *Natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ... References External links dinatural transformationat the ''n''-Category Lab. Functors {{categorytheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
