nLab
two-variable adjunction

Two-variable and n-variable adjunctions

Idea

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

by extracting the central pattern.

Definition

Let C, D and E 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(D,hom l(C,E))Hom D(C,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 (,hom l,hom r):C×DE 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}

and

(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 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

Revised on September 12, 2012 03:22:51 by Mike Shulman (71.136.244.71)