### Context

category theory

#### 2-Category theory

2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

# Two-variable and $n$-variable adjunctions

## Idea

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

1. $V$-enriched categories having powers and copowers.

## 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.

## References

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

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

The terminology adjunction of two variables is used in:

and:

Generalization to $n$-variable adjunctions:

Last revised on December 16, 2023 at 17:13:21. See the history of this page for a list of all contributions to it.