2Category

A 2-category is a category enriched over Cat. It has objects, 1-morphisms between objects, and 2-morphisms between 1-morphisms. For any pair of objects A and B, the collection Hom(A, B) is itself a category. Its objects are 1-morphisms f: A → B, and its morphisms are 2-morphisms α: f ⇒ g between such 1-morphisms.

There is a composition operation that assigns to f: A → B and g: B → C a composite g ∘ f: A → C, and this induces a functor Hom(B, C) × Hom(A, B) → Hom(A, C) giving horizontal composition of 1-morphisms. Each object A has an identity 1-morphism id_A: A → A, which acts as a unit for composition. In a 2-category, the associativity of composition and the unit laws hold strictly. 2-morphisms can be composed vertically within a Hom(A, B) category, and there is horizontal composition of 2-morphisms compatible with the 1-morphism composition. The interchange law governs how vertical and horizontal compositions interact.

Examples include the canonical 2-category Cat, whose objects are (small) categories, 1-morphisms are functors, and 2-morphisms are natural transformations. Other instances arise from suitable collections of mathematical structures where morphisms themselves carry morphisms between them, such as certain 2-categories built from spans, profunctors, or more generally enriched categorical contexts.

Relation to bicategories: Every 2-category is a bicategory with strict associativity and unit laws, but a bicategory only requires associativity and unit laws to hold up to specified isomorphisms, subject to coherence conditions. Thus bicategories generalize 2-categories by weakening these equalities.