Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
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.
Given a pair of strict 2-functors between strict 2-categories
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
which is strict 2-natural in $a$ and $c$;
a pair of strict 2-natural transformations of 2-functors:
adjunction unit$\; \eta \colon Id_A \to U F$,
adjunction counit$\; \epsilon \colon F U \to Id_B$,
satisfying the triangle identities strictly.
For more general notions of 2-adjunctions, involving weak 2-categories, weak 2-functors, and/or weak 2-natural transformations; see at 2-adjunction.
Last revised on February 12, 2024 at 15:36:22. See the history of this page for a list of all contributions to it.