Context

2-Category theory

2-category theory

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

$\left(\otimes ,{\mathrm{hom}}_{l},{\mathrm{hom}}_{r}\right):C×D\to E$(\otimes, hom_l, hom_r) : C \times D \to E

consists of bifunctors

$\begin{array}{rl}\otimes & :C×D\to E\\ {\mathrm{hom}}_{l}& :{C}^{\mathrm{op}}×E\to D\\ {\mathrm{hom}}_{r}& :{D}^{\mathrm{op}}×E\to C\end{array}$\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

${\mathrm{Hom}}_{E}\left(C\otimes D,E\right)\simeq {\mathrm{Hom}}_{C}\left(D,{\mathrm{hom}}_{l}\left(C,E\right)\right)\simeq {\mathrm{Hom}}_{D}\left(C,{\mathrm{hom}}_{r}\left(D,E\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$Hom_E(C \otimes D, E) \simeq Hom_C(D, hom_l(C,E)) \simeq Hom_D(C, hom_r(D,E)) \,.

Cyclicity

If $\left(\otimes ,{\mathrm{hom}}_{l},{\mathrm{hom}}_{r}\right):C×D\to E$ is a two-variable adjunction, then so are

$\left({\mathrm{hom}}_{l}^{\mathrm{op}},{\otimes }^{\mathrm{op}},{\mathrm{hom}}_{r}\right):{E}^{\mathrm{op}}×C\to {D}^{\mathrm{op}}$(hom_l^{op}, \otimes^{op}, hom_r) : E^{op} \times C \to D^{op}

and

$\left({\mathrm{hom}}_{r}^{\mathrm{op}},{\mathrm{hom}}_{l},{\otimes }^{\mathrm{op}}\right):D×{E}^{\mathrm{op}}\to {C}^{\mathrm{op}}.$(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” $C×D\to {E}^{\mathrm{op}}$; 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)