nLab dual adjunction

Idea

A dual adjunction between categories $C$ and $D$ is an adjunction between the opposite category $C^{op}$ of $C$ and $D$.

The concept arises in the context of duality.

Dual adjunctions between concrete categories are frequently represented by dualizing objects.

Dual adjunctions between posets are also called Galois connections.

Definition

A dual adjunction consists of contravariant functors $F: C \to D$, $G: D \to C$ together with natural transformations $\eta: 1_C \to G F$ and $\theta: 1_D \to F G$ such that $F \eta \circ \theta F = 1_F$ and $G \theta \circ \eta G = 1_G$. In diagrams, the following must commute.

Reformulated in terms of covariant functors, a dual adjunction can be viewed as an ordinary adjunction $F \dashv G$ with $F: C \to D^{op}$ and $G: D^{op} \to C$, or as $G \dashv F$ with $G: D \to C^{op}$ and $F: C^{op} \to D$. However, it is often useful not to break the symmetry of the contravariant formulation.

A self-dual adjunction is a dual adjunction for which $F = G: C \to C$ and $\eta = \theta: 1 \to F F$. An example is where $C$ is a symmetric monoidal closed category and $F = [-, d]$ is an internal hom into an object $d$, where the unit is the usual double-dual embedding $\delta_c: c \to [[c, d], d]$.

Related concepts

• adjunction

  • strong adjoint functor

  • adjoint triple, adjoint quadruple, recollement

  • ambidextrous adjunction

  • adjoint monad

  • fixed point of an adjunction

  • enriched adjunction

  • two-variable adjunction

References

The concept appears by the name of “mutually right adjoint functors” and “mutually left adjoint functors” in Definition 1.3 of:

• Emily Riehl, Multivariable adjunctions and mates, J. K-Theory 13 (2014), 337–396, doi:10.1017/is013012007jkt250, arXiv:1208.4520.

and in Definition 4.4.1 of:

• Emily Riehl, Category Theory in Context, upcoming 2nd ed. preliminary pdf