|
2-category
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 compo ... [...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] |
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] |
Enriched Category
In category theory, a branch of mathematics, an enriched category generalizes the idea of a category (mathematics), category by replacing hom-sets with objects from a general monoidal category. It is motivated by the observation that, in many practical applications, the hom-set often has additional structure that should be respected, e.g., that of being a vector space of morphisms, or a topological space of morphisms. In an enriched category, the set of morphisms (the hom-set) associated with every pair of objects is replaced by an object (category theory), object in some fixed monoidal category of "hom-objects". In order to emulate the (associative) composition of morphisms in an ordinary category, the hom-category must have a means of composing hom-objects in an associative manner: that is, there must be a binary operation on objects giving us at least the structure of a monoidal category, though in some contexts the operation may also need to be commutative and perhaps also to ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enriched Categories
In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category. It is motivated by the observation that, in many practical applications, the hom-set often has additional structure that should be respected, e.g., that of being a vector space of morphisms, or a topological space of morphisms. In an enriched category, the set of morphisms (the hom-set) associated with every pair of objects is replaced by an object in some fixed monoidal category of "hom-objects". In order to emulate the (associative) composition of morphisms in an ordinary category, the hom-category must have a means of composing hom-objects in an associative manner: that is, there must be a binary operation on objects giving us at least the structure of a monoidal category, though in some contexts the operation may also need to be commutative and perhaps also to have a right adjoint (i.e., making the category ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherent Isomorphism
In mathematics, specifically in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or "up to isomorphism". The adjectives such as "pseudo-" and "lax-" are used to refer to the fact equalities are weakened in coherent ways; e.g., pseudo-functor, pseudoalgebra. Coherent isomorphism In some situations, isomorphisms need to be chosen in a coherent way. Often, this can be achieved by choosing canonical isomorphisms. But in some cases, such as prestacks, there can be several canonical isomorphisms and there might not be an obvious choice among them. In practice, coherent isomorphisms arise by weakening equalities; e.g., strict associativity may be replaced by associativity via coherent isomorphisms. For example, via this process, one gets the notion of a weak 2-category from that of a strict 2-category. Replacing coherent isomorphisms by equalities is usually called strictification o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tricategory
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] |
Charles Ehresmann
Charles Ehresmann (19 April 1905 – 22 September 1979) was a German-born French mathematician who worked in differential topology and category theory. He was an early member of the Bourbaki group, and is known for his work on the differential geometry of smooth fiber bundles, notably the introduction of the concepts of Ehresmann connection and of jet bundles, and for his seminar on category theory. Life Ehresmann was born in Strasbourg (at the time part of the German Empire) to an Alsatian-speaking family; his father was a gardener. After World War I, Alsace returned part of France and Ehresmann was taught in French at Lycée Kléber. Between 1924 and 1927 he studied at the École Normale Supérieure (ENS) in Paris and obtained agrégation in mathematics. After one year of military service, in 1928-29 he taught at a French school in Rabat, Morocco. He studied further at the University of Göttingen during the years 1930–31, and at Princeton University in 1932–34. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoidal Category
In mathematics, a monoidal category (or tensor category) is a category (mathematics), category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an Object (category theory), object ''I'' that is both a left identity, left and right identity for ⊗, again up to a natural isomorphism. The associated natural isomorphisms are subject to certain coherence conditions, which ensure that all the relevant diagram (category theory), diagrams commutative diagram, commute. The ordinary tensor product makes vector spaces, abelian groups, module (mathematics), ''R''-modules, or algebra (ring theory), ''R''-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples. Every (small category, small) monoidal category may also be viewed as a "categorification" of an underlying monoid, namely the monoid whose elements are the isomorphism classes of the category ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Small Categories
In mathematics, specifically in category theory, the category of small categories, denoted by Cat, is the category whose objects are all small categories and whose morphisms are functors between categories. Cat may actually be regarded as a 2-category with natural transformations serving as 2-morphisms. The initial object of Cat is the ''empty category'' 0, which is the category of no objects and no morphisms. The terminal object is the ''terminal category'' or ''trivial category'' 1 with a single object and morphism. at nLab The category Cat is itself a large category, and therefore not an object of itself. In order to avoid problems analogous to [...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] |
Class (set Theory)
In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. Classes act as a way to have set-like collections while differing from sets so as to avoid paradoxes, especially Russell's paradox (see '). The precise definition of "class" depends on foundational context. In work on Zermelo–Fraenkel set theory, the notion of class is informal, whereas other set theories, such as von Neumann–Bernays–Gödel set theory, axiomatize the notion of "proper class", e.g., as entities that are not members of another entity. A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class, and a class that is a set is sometimes called a small class. For instance, the class of all ordinal numbers, and the class of all sets, are proper classes in many formal systems. In Quine's set-theoretical writing, the phrase "ultimate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is derived . The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may often be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism. A common example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension one of a vector space are isomorphic and cannot be identified. An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
