Bicategory

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:

• 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 1-cells p, q is referred to as vertical composition and is denoted p·q.
• 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

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.

See also: Bicategory, Associativity, Category of small categories, Category theory, Final object, Functor, Isomorphism, Map (mathematics), Mathematics