nLab strict adjoint 2-functor




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 𝒱 Cat \mathcal{V}Cat , for 𝒞=\mathcal{C} = Cat.

The following indicates what this means in more explicit detail.


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

F:AC:U F \,\colon\, A \rightleftarrows C \,\colon\, U

a strict 2-adjunction between them is, equivalently:

* a pair of [[strict 2-natural transformations</a> of 2-functors:

  1. adjunction unitη:Id AUF\eta \colon Id_A \to U F,

  2. adjunction counitϵ:FUId B\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 May 18, 2023 at 13:39:40. See the history of this page for a list of all contributions to it.