2-functor

Definition

A 2-functor is a Functor between 2-categories that preserves the 2-categorical structure, including objects, 1-morphisms, and 2-morphisms.

Given 2-categories $\mathcal{C}$ and $\mathcal{D}$, a 2-functor $F:\mathcal{C}\to \mathcal{D}$ consists of:

• A mapping of objects: $F:\text{Ob}(\mathcal{C})\to \text{Ob}(\mathcal{D})$
• For each pair of objects $A,B$ in $\mathcal{C}$, a functor $F_{A,B}:\mathcal{C}(A,B)\to \mathcal{D}(F(A),F(B))$ between hom-categories
• Preservation of composition and identities at both the 1-morphism and 2-morphism levels

Types

2-functors can be:

• Strict: Preserve composition and identities exactly (on the nose)
• Weak (pseudofunctors): Preserve composition and identities up to coherent natural isomorphisms

Related Concepts

• Functor: The 1-categorical analogue
• 2-Category: The domain and codomain of 2-functors
• Natural Transformation: 2-morphisms between functors
• Higher Category Theory: Generalizations to n-functors