nLab two-variable adjunction

Two-variable and -variable adjunctions

Two-variable and nn-variable adjunctions


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

  1. internal homs in a biclosed monoidal category

  2. VV-enriched categories having powers and copowers.


Let CC, DD and EE be categories. An adjunction of two variables or two-variable adjunction

(,hom l,hom r):C×DE (\otimes, hom_l, hom_r) \,\colon\, C \times D \longrightarrow E

consists of bifunctors of this form

:C×DE hom l :C op×ED hom r :D op×EC \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:

cC,dD,eEE(cd,e)D(d,hom l(c,e))C(c,hom r(d,e)). 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) \,.


If (,hom l,hom r):C×DE(\otimes, hom_l, hom_r) : C \times D \to E is a two-variable adjunction, then so are

(hom l op, op,hom r):E op×CD op (hom_l^{op}, \otimes^{op}, hom_r) : E^{op} \times C \to D^{op}


(hom r op,hom l, op):D×E opC 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×DE opC\times D \to E^{op}; see Cheng-Gurski-Riehl.

Adjunctions of nn variables

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


Under the name “adjunction with a parameter” the concept appears in:

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

The terminology adjunction of two variables is used in:

Generalization to nn-variable adjunctions:

Last revised on May 10, 2023 at 15:34:01. See the history of this page for a list of all contributions to it.