nLab two-variable adjunction

Two-variable and -variable adjunctions

Two-variable and nn-variable adjunctions


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


by extracting the central pattern.


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) : C \times D \to E

consists of bifunctors

:C×DE hom l :C op×ED hom r :D op×EC \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(cd,e)D(d,hom l(c,e))C(c,hom r(d,e)) E(c \otimes d, e) \simeq D(d, hom_l(c,e)) \simeq C(c, hom_r(d,e))


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.

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.


  • 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

Last revised on December 1, 2022 at 17:29:50. See the history of this page for a list of all contributions to it.