Bicategory
A bicategory or weak
-category is a tuple consisting of
- A category
; - Hom Categories: for each pair of objects (
-cells) , is equipped with a category , called a hom category.
- Its objects are called
-cells in . The collection of all the -cells ( -morphisms) in is denoted by . - Its morphisms are called
-cells in . The collection of all the -cells in 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 . - Identity
-Cells: For each object , denote as the singleton category, is a functor. We identify the functor 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
-cells , , and :
- Unity Axiom: The middle unity diagram commutes:
- Pentagon Axiom:
We usually abbreviate a bicategory as above to
.
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.