nLab two-variable adjunction

Idea

An adjunction of two variables is a straightforward generalization of both:

• the internal hom in a biclosed monoidal category

and

• a $V$-enriched category having powers and copowers

by extracting the central pattern.

Definition

Let $C$, $D$ and $E$ be categories. An adjunction of two variables or two-variable adjunction

$$(\otimes, hom_l, hom_r) : C \times D \to E$$

consists of bifunctors

$$\begin{aligned} \otimes & : C \times D \to E \\ hom_l &: C^{op} \times E \to D \\ hom_r &: D^{op} \times E \to C \end{aligned}$$

together with natural isomorphisms

$$E(c \otimes d, e) \simeq D(d, hom_l(c,e)) \simeq C(c, hom_r(d,e))$$

Cyclicity

If $(\otimes, hom_l, hom_r) : C \times D \to E$ is a two-variable adjunction, then so are

$$(hom_l^{op}, \otimes^{op}, hom_r) : E^{op} \times C \to D^{op}$$

and

$$(hom_r^{op}, hom_l, \otimes^{op}) : D\times E^{op} \to C^{op}.$$

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” $C\times D \to E^{op}$; see Cheng-Gurski-Riehl.

Adjunctions of $n$ variables

There is a straightforward generalization to an adjunction of $n$ variables, which involves $n+1$ categories and $n+1$ functors. Adjunctions of $n$ 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.

Adjunctions with a parameter

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.

Related concepts

• Quillen bifunctor

References

• 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