Contents

Idea

A strict 2-functor is a 2-functor between strict 2-categories which strictly respects the composition operation of 1-morphisms and 2-morphisms. This is in contrast to weak 2-functors (also called pseudofunctors) which may respect the composition of 1-morphisms only up to natural isomorphism (hence up to 2-morphisms in the 2-category of categories).

While the generally “correct” concepts in 2-category theory are weak 2-categories with weak 2-functors between them, it is often useful to recognize strict 2-functors among weak 2-functors, or to replace, up to eqivalence of 2-categories and pseudonatural transformation , weak 2-functors by strict ones.

Definition

Let Cat be the 1-category of categories, regarded as a Bénabou cosmos for enriched category theory via its cartesian closed category-structure.

Notice that strict 2-categories may be identified with Cat-enriched categories. Under this identification, strict 2-functors are Cat-enriched functors.

Related concepts

• strict 2-category

• strict 2-groupoid

• strict 2-group

• strict 2-natural transformation

• strict adjoint 2-functor

• 2-functor, double functor

• n-functor