nLab two-variable adjunction

Idea

The notion of adjunction of two variables is a natural generalization of both that of:

1. internal homs in a biclosed monoidal category

2. $V$-enriched categories having powers and copowers (see powered and copowered category)

Definition

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

$$(\otimes, hom_l, hom_r) \,\colon\, C \times D \longrightarrow E$$

consists of bifunctors of this form

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

together with natural isomorphism of this form:

$$c \in C,\, d \in D,\, e \in E \;\;\;\;\;\;\; \vdash \;\;\;\;\;\;\; E\big(c \otimes d,\, e\big) \;\simeq\; D\big(d,\, hom_l(c,e)\big) \;\simeq\; C\big(c,\, hom_r(d,e)\big) \,.$$

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.

Related concepts

• adjoint functor

• Quillen bifunctor

• parameterized monad

References

Under the name “adjunction with a parameter” the concept appears in p. 102 of:

• Saunders MacLane, §IV.7, Thm. 3 (p. 100) of: Categories for the working mathematician, Graduate Texts in Mathematics, Springer (1971) [doi:10.1007/978-1-4757-4721-8]

Under the name “THC-situation” the concept is discussed in:

• John Gray, Def. 1.1 in: Closed categories, lax limits and homotopy limits. J. Pure Appl. Algebra 19 (1980) 127 - 158 [doi:10.1016/0022-4049(80)90098-5]

The terminology adjunction of two variables is used in:

• Mark Hovey, Def. 4.1.12 in: Model Categories, Mathematical Surveys and Monographs, 63 AMS (1999) [ISBN:978-0-8218-4361-1, doi:10.1090/surv/063, pdf, Google books]

and:

• Emily Riehl, Def. 4.3.7 in: Category Theory in Context, Dover Publications (2017)

Generalization to $n$-variable adjunctions:

• Eugenia Cheng, Nick Gurski, Emily Riehl, “Multivariable adjunctions and mates” [arXiv:1208.4520]

• Rene Guitart, Trijunctions and triadic Galois connections. Cah. Topol. Géom. Différ. Catég. 54 (2013), no. 1, 13-27, (pdf)

• Mike Shulman, The 2-Chu-Dialectica construction and the polycategory of multivariable adjunctions, arxiv:1806.06082, blog post