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2-functor
In mathematics, a 2-functor is a morphism between 2-categories. They may be defined formally using enrichment Enrichment may refer to: * Behavioral enrichment, the practice of providing animals under managed care with stimuli such as natural and artificial objects * Data enrichment, appending or enhancing data with relevant context from other sources, se ... by saying that a 2-category is exactly a ''Cat''-enriched category and a 2-functor is a ''Cat''-functor. Explicitly, if ''C'' and ''D'' are 2-categories then a 2-functor F\colon C\to D consists of * a function F\colon \text C\to \text D, and * for each pair of objects c,c'\in C a functor F_\colon \text_(c,c')\to\text_D(Fc,Fc') such that each F_ strictly preserves identity objects and they commute with horizontal composition in ''C'' and ''D''. See for more details and for lax versions. References Functors Higher category theory {{categorytheory-stub ... [...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 poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2-category
In category theory, a strict 2-category is a category (mathematics), 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 category, enriched over Cat (the Category of small categories, category of categories and functors, with the monoidal category, monoidal structure given by product category, 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 (set theory), class of 0-''cells'' (or ''Object (category theory), objects'') , , . ... [...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 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 catego ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lax Functor
In category theory, a discipline within mathematics, the notion of lax functor between bicategories generalizes that of functors between categories. Let ''C,D'' be bicategories. We denote composition idiagrammatic order A ''lax functor P from C to D'', denoted P: C\to D, consists of the following data: * for each object ''x'' in ''C'', an object P_x\in D; * for each pair of objects ''x,y ∈ C'' a functor on morphism-categories, P_: C(x,y)\to D(P_x,P_y); * for each object ''x∈C'', a 2-morphism P_:\text_\to P_(\text_x) in ''D''; * for each triple of objects, ''x,y,z ∈C'', a 2-morphism P_(f,g): P_(f);P_(g)\to P_(f;g) in ''D'' that is natural in ''f: x→y'' and ''g: y→z''. These must satisfy three commutative diagrams, which record the interaction between left unity, right unity, and associativity between ''C'' and ''D''. See http://ncatlab.org/nlab/show/pseudofunctor. A lax functor in which all of the structure 2-morphisms, i.e. the P_ and P_{x,y,z} above, are invertible ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functors
In mathematics, specifically category theory, a functor is a mapping between 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 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 linguistic context; see function word. Definition Let ''C'' and ''D'' be 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'', * associates each morphism f \colon X \to Y in ''C'' to a morphism F(f) \colon F(X) \to F(Y) in ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
