Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
An adjunction of two variables is a straightforward generalization of both:
and
by extracting the central pattern.
Let , and be categories. An adjunction of two variables or two-variable adjunction
consists of bifunctors
together with natural isomorphisms
If is a two-variable adjunction, then so are
and
giving an action of the cyclic group of order 3. This can be made to look more symmetrical by regarding the original two-variable adjunction as a “two-variable left adjunction” ; see Cheng-Gurski-Riehl.
There is a straightforward generalization to an adjunction of variables, which involves categories and functors. Adjunctions of variables assemble into a 2-multicategory. They also have a corresponding notion of mates; see Cheng-Gurski-Riehl.
This 2-multicategory can also be promoted to a 2-polycategory; see Shulman.
In CWM Theorem IV.7.3, Mac Lane introduced the notion of adjunction with a parameter, which is equivalent to the notion of two-variable adjunction.
John Gray, Closed categories, lax limits and homotopy limits. J. Pure Appl. Algebra 19 (1980), 127–158. Possibly the oldest abstract definition of the concept, under the name “THC-situation”.
Mark Hovey. Model Categories, volume 63 of Mathematical Surveys and Monographs. American Mathematical Society, 1999. See Chapter 4.
Rene Guitart, Trijunctions and triadic Galois connections. Cah. Topol. Géom. Différ. Catég. 54 (2013), no. 1, 13–27.
Eugenia Cheng, Nick Gurski, Emily Riehl, “Multivariable adjunctions and mates”, arXiv:1208.4520.
Mike Shulman, The 2-Chu-Dialectica construction and the polycategory of multivariable adjunctions, arxiv:1806.06082, blog post
Last revised on December 1, 2022 at 17:29:50. See the history of this page for a list of all contributions to it.