nLab strict adjoint 2-functor

Idea

This entry means to indicate a generalization of the notion of adjoint functors from category theory to 2-category theory, but specialized to the case of strict 2-categories.

Notice that strict 2-categories may be understood as Cat-enriched categories, for Cat understood as the 1-category of small strict categories with functors between them and equipped with its cartesian monoidal-structure (via forming product categories).

In this sense the respective notion of adjoint 2-functors between strict 2-categories is that of Cat-enriched adjoint functors, which (by the discussion there) is equivalently that of adjunctions in the 2-category$\mathcal{V}Cat$, for $\mathcal{C} =$ Cat.

The following indicates what this means in more explicit detail.

Definition

Given a pair of strict 2-functors between strict 2-categories

$$F \,\colon\, A \rightleftarrows C \,\colon\, U$$

a strict 2-adjunction between them is, equivalently:

• for each pair of objects $a \in A$, $c\in C$, a natural isomorphism of strict hom-categories

  $$C\big(F(a), c\big) \,\cong\, A\big(a, U(c)\big)$$

  which is strict 2-natural in $a$ and $c$;

• a pair of strict 2-natural transformations of 2-functors:

  1. adjunction unit$\; \eta \colon Id_A \to U F$,

  2. adjunction counit$\; \epsilon \colon F U \to Id_B$,

  satisfying the triangle identities strictly.

Related concepts

For more general notions of 2-adjunctions, involving weak 2-categories, weak 2-functors, and/or weak 2-natural transformations; see at 2-adjunction.