Bicategory

From Academic Kids

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 IX : 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)
    • aW,X,Y,Z : ;W,X,Z ° (Id × ;X,Y,Z) → ;W,Y,Z ° (;W,X,Y × Id)
    • rX,Y : ;X,X,Y ° (IX × Id) → Id
    • lX,Y : ;X,Y,Y ° (Id × IY) → Id
[edit]

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.Template:Math-stub

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