Bicategory

In mathematics, a bicategory (or a weak 2-category) is a concept in category theory used to extend the notion of category to handle the cases where the composition of morphisms is not (strictly) associative, but only associative "up to" an isomorphism. The notion was introduced in 1967 by Jean Bénabou.
Bicategories may be considered as a weakening of the definition of 2-categories. A similar process for 3-categories leads to tricategories, and more generally to weak "n"-categories for "n"-categories.
Definition.
Formally, a bicategory B consists of:
with some more structure:
The horizontal composition is required to be associative up to a natural isomorphism α between morphisms formula_2 and formula_3. Some more coherence axioms, similar to those needed for monoidal categories, are moreover required to hold: a monoidal category is the same as a bicategory with one 0-cell.