duality

## In QFT and String theory

#### 2-Category theory

2-category theory

## 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 $G \theta \circ \eta G = 1_G$.

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 $\delta_c: c \to [[c, d], d]$.

Revised on January 25, 2015 19:00:20 by Todd Trimble (67.81.95.215)