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Modification (mathematics)
In mathematics, specifically category theory, a modification is an arrow between 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 ...s. It is a 3-cell in the 3-category of 2-cells (where the 2-cells are natural transformations, the 1-cells are functors, and the 0-cells are categories). The notion is due to Bénabou. Given two natural transformations \boldsymbol : \boldsymbol \rightarrow \boldsymbol, there exists a modification \boldsymbol : \boldsymbol \rightarrow \boldsymbol such that: * \boldsymbol : \boldsymbol \rightarrow \boldsymbol, * \boldsymbol : \boldsymbol \rightarrow \boldsymbol, and * \boldsymbol : \boldsymbol \rightarrow \boldsymbol . The following commutative diagram shows an example of a modification and its inner workings. References *{ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in most areas of mathematics. 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 space (other), quotient spaces, direct products, completion, and duality (mathematics), duality. Many areas of computer science also rely on category theory, such as functional programming and Semantics (computer science), semantics. A category (mathematics), category is formed by two sorts of mathematical object, objects: the object (category theory), objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. Metapho ... [...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 (both from C to 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higher Category Theory
In mathematics, higher category theory is the part of category theory at a ''higher order'', which means that some equalities are replaced by explicit morphism, arrows in order to be able to explicitly study the structure behind those equalities. Higher category theory is often applied in algebraic topology (especially in homotopy theory), where one studies algebraic Invariant (mathematics), invariants of topological space, spaces, such as the Fundamental groupoid, fundamental . In higher category theory, the concept of higher categorical structures, such as (), allows for a more robust treatment of homotopy theory, enabling one to capture finer homotopical distinctions, such as differentiating two topological spaces that have the same fundamental group but differ in their higher homotopy groups. This approach is particularly valuable when dealing with spaces with intricate topological features, such as the Eilenberg-MacLane space. Strict higher categories An ordinary category (m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-category
In mathematics, especially in category theory, a 3-category is a 2-category together with 3-morphisms. It comes in at least three flavors *a strict 3-category, *a semi-strict 3-category also called a Gray category, *a weak 3-category. The coherence theorem of Gordon–Power–Street says a weak 3-category is equivalent (in some sense) to a Gray category. Strict and weak 3-categories A strict 3-category is defined as a category enriched over 2Cat, the monoidal category of (small) strict 2-categories. A weak 3-category is then defined roughly by replacing the equalities in the axioms by coherent isomorphisms. Gray tensor product Introduced by Gray, a Gray tensor product is a replacement of a product of 2-categories that is more convenient for higher category theory. Precisely, given a morphism f : x \to y in a strict 2-category ''C'' and g:a \to b in ''D'', the usual product is given as f \times g : (x, a) \to (y, b) that factors both as u = (\operatorname, g) \circ (f, \operat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2-morphism
In category theory in mathematics, a 2-category is a category with "morphisms between morphisms", called 2-morphisms. A basic example is the category Cat of all (small) categories, where a 2-morphism is a natural transformation between functors. The concept of a strict 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 1967 by Jean Bénabou. A (2, 1)-category is a 2-category where each 2-morphism is invertible. Definitions A strict 2-category By definition, a strict 2-category ''C'' consists of the data: * a class of 0-''cells'', * for each pairs of 0-cells a, b, a set \operatorname(a, b) called the set of 1-''cells'' from a to b, * for each pairs of 1-cells f, g in the same hom-set, a set \operatorname(f, g) called the set of 2-''cells'' from f to g, * ''ordinary composi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean Bénabou
Jean Bénabou (1932 – 11 February 2022) was a French mathematician, known for his contributions to category theory. He directed the Research Seminar in Category Theory at the Institut Henri Poincaré The Henri Poincaré Institute (or IHP for ''Institut Henri Poincaré'') is a mathematics research institute part of Sorbonne University, in association with the Centre national de la recherche scientifique (CNRS). It is located in the 5th arrondi ... and Institut de mathématiques de Jussieu from 1969 to 2001. Partial bibliography * * * * * * References External links * 1932 births 2022 deaths 20th-century French mathematicians Category theorists Moroccan emigrants to France People from Rabat University of Paris alumni External links * {{France-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modification In Category Theory
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Modification may refer to: * Modifications of school work for students with special educational needs * Modifications (genetics), changes in appearance arising from changes in the environment * Posttranslational modifications, changes to proteins arising from protein biosynthesis * Modding, modifying hardware or software ** Mod (video gaming) ** Modified car * Body modification * Grammatical modifier * Home modifications * Chemical modification, processes involving the alteration of the chemical constitution or structure of molecules See also * * Modified (other) * Modifier (other) * Mod (other) * Edit (other) * Manipulation (other) Manipulation may refer to: * Manipulation (psychology) - acts intended to influence or control someone in a underhanded or subtle way * Crowd manipulation - use of crowd psychology to direct the behavior of a crowd toward a specific action * Inte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in most areas of mathematics. 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 space (other), quotient spaces, direct products, completion, and duality (mathematics), duality. Many areas of computer science also rely on category theory, such as functional programming and Semantics (computer science), semantics. A category (mathematics), category is formed by two sorts of mathematical object, objects: the object (category theory), objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. Metapho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
