two-variable adjunction

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

Hom E(cd,e)Hom C(c,hom l(d,e))Hom D(d,hom r(c,e)). Hom_E(c \otimes d, e) \simeq Hom_C(c, hom_l(d,e)) \simeq Hom_D(d, hom_r(c,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.


Revised on April 20, 2017 04:31:06 by Mike Shulman (