2-category

In <a href="/facts/Category_theory/6zS3DXPa">category theory</a> in mathematics, a 2-category is a <a href="/facts/Category_(mathematics)/7WzxUThf">category</a> with "<a href="/facts/Morphism/Kdpuyz3S">morphisms</a> between morphisms", called 2-morphisms. A basic example is the category Cat of all (small) categories, where a 2-morphism is a <a href="/facts/Natural_transformation/xwPumTHt">natural transformation</a> between <a href="/facts/Functor/l2u3rIBE">functors</a>.
The concept of a strict 2-category was first introduced by <a href="/facts/Charles_Ehresmann/KNtePSlk">Charles Ehresmann</a> in his work on <a href="/facts/Enriched_categories/VgphwkBu">enriched categories</a> in 1965. The more general concept of bicategory (or weak 2-category), where composition of morphisms is <a href="/facts/Associative/EQX8lV6r">associative</a> only up to a 2-isomorphism, was introduced in 1967 by <a href="/facts/Jean_B%25C3%25A9nabou/mQlt5gFm">Jean Bénabou</a>.
A (2, 1)-category is a 2-category where each 2-morphism is invertible.

2-category open-in-new

2-category