In mathematics, a bicategory is a concept in category theory used to extend the notion of sameness (i.e. isomorphism) to the morphisms of a category. A bicategory B consists of the following: Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...

• A set of objects X, Y, Z, ... called 0-cells.
• Between every two objects X and Y, a set of morphisms p, q, r, ... denoted B(X,Y) called 1-cells. In a bicategory these 1-cells form a small category themselves by introducing mappings s, t, u, ... between 1-cells p, q ∈ B(X,Y). These maps s, t, u, etc. are referred to as 2-cells. Composition of 2-cells s, t (both between 1-cells in B(X,Y)) is referred to as vertical composition and is denoted s·t.
• For every three 0-cells X, Y, Z, a (bi)functor ;_X,Y,Z : B(X,Y) × B(Y,Z) → B(X,Z), which provides for horizontal composition. The associativity and unit laws for ; are relaxed from the usual equality to holding up to an isomorphism. The two types of composition (horizontal and vertical) follow the equation (s·t);(u·v) = (s;t)·(u;v) for 2-cells s,t,u,v.
• For every 0-cell X, a functor I_X : 1 → B(X,X) where 1 denotes the final object in the category Cat of small categories (the category with small categories as objects and the functors between them as morphisms).
• For 0-cells W,X,Y,Z, natural isomorphisms (with Id denoting the identity functor and ° functor composition)
  • a_W,X,Y,Z : ;_W,X,Z ° (Id × ;_X,Y,Z) → ;_W,Y,Z ° (;_W,X,Y × Id)
  • r_X,Y : ;_X,X,Y ° (I_X × Id) → Id
  • l_X,Y : ;_X,Y,Y ° (Id × I_Y) → Id

In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics and related technical fields, the term map or mapping is often a synonym for function. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... In mathematics, associativity is a property that a binary operation can have. ... Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...

Examples

The category of small categories, Cat, forms a bicategory with small categories as 0-cells, functors as 1-cells, and natural transformations as 2-cells. In mathematics, specifically in category theory, the 2-category of small categories is the 2-category whose objects are small categories, whose arrows are functors and whose 2-arrows are natural transformations. ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...