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

$$(\otimes ,{\mathrm{hom}}_l,{\mathrm{hom}}_r):C\times D\to E$$(\otimes, hom_l, hom_r) : C \times D \to E

consists of bifunctors

$$\begin{array}{rl}\otimes & :C\times D\to E\\ {\mathrm{hom}}_l& :C^{\mathrm{op}}\times E\to D\\ {\mathrm{hom}}_r& :D^{\mathrm{op}}\times E\to C\end{array}$$\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

$${\mathrm{Hom}}_E(C\otimes D,E)\simeq {\mathrm{Hom}}_C(D,{\mathrm{hom}}_l(C,E))\simeq {\mathrm{Hom}}_D(C,{\mathrm{hom}}_r(D,E)).$$Hom_E(C \otimes D, E) \simeq Hom_C(D, hom_l(C,E)) \simeq Hom_D(C, hom_r(D,E)) \,.

Cyclicity

If $(\otimes ,{\mathrm{hom}}_l,{\mathrm{hom}}_r):C\times D\to E$ is a two-variable adjunction, then so are

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

and

$$({\mathrm{hom}}_r^{\mathrm{op}},{\mathrm{hom}}_l,{\otimes }^{\mathrm{op}}):D\times E^{\mathrm{op}}\to C^{\mathrm{op}}.$$(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” $C\times D\to E^{\mathrm{op}}$; 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.

References

• Mark Hovey. Model Categories, volume 63 of Mathematical Surveys and Monographs. American Mathematical Society, 1999. See Chapter 4.

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