Bicategories

The idea of a bicategory comes from natural transformations being the morphisms between functors,

Definition of Bicategory

Bicategory

A bicategory or weak -category is a tuple consisting of

• A collection of objects (called -cells) .
• Hom Categories: for each pair of -cells , it is equipped with a category , called a hom category.
  • Its objects are called -cells. The collection of all the -cells is denoted by .
  • Its morphisms are called -cells. The collection of all the -cells is denoted by .
  • Composition law and identity morphisms in the category are called vertical composition and identity -cells, respectively.
  • An isomorphism in is called an invertible -cell, and its inverse is called a vertical inverse. For a -cell , its identity -cell is denoted by .
  • For -cells , we display each -cell in diagrams as:
    
• Identity -Cells: For each object , denote as the singleton category, there is a distinguished functor We identify it with the -cell , called the identity -cell of .
• Horizontal Composition: For each triple of objects , is a functor, called the horizontal composition. For -cells and , and -cells and , we use the notations
• Associator: For objects , is a natural isomorphism, called the associator, between functors
• Unitors: For each pair of objects , are natural isomorphisms, called the left unitor and the right unitor, respectively.

And the following axioms hold for all -cells , , and :

• Unity Axiom: The middle unity diagram in commutes:
• Pentagon Axiom: The following diagram in commutes:

We usually abbreviate a bicategory as above to .

e.g.

• A monoidal category may be regarded a bicategory with a single object , where it serves as the hom category , and the horizontal composition is given by the monoidal product .

Definition

Def Local Property
Suppose is a property of categories. A bicategory is locally if each hom category in has property . In particular, is:

• locally small if each hom category is a small category.
• locally discrete if each hom category is discrete.
• locally partially ordered if each hom category is a partially ordered set regarded as a small category.