nLab
dual adjunction

Context

Duality

2-Category theory

Idea

A dual adjunction between categories CC and DD is an adjunction between the opposite category C opC^{op} of CC and DD.

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:CDF: C \to D, G:DCG: D \to C together with natural transformations η:1 CGF\eta: 1_C \to G F and θ:1 DFG\theta: 1_D \to F G such that FηθF=1 FF \eta \circ \theta F = 1_F and GθηG=1 GG \theta \circ \eta G = 1_G.

Reformulated in terms of covariant functors, a dual adjunction can be viewed as an ordinary adjunction FGF \dashv G with F:CD opF: C \to D^{op} and G:D opCG: D^{op} \to C, or as GFG \dashv F with G:DC opG: D \to C^{op} and F:C opDF: 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:CCF = G: C \to C and η=θ:1FF\eta = \theta: 1 \to F F. An example is where CC is a symmetric monoidal closed category and F=[,d]F = [-, d] is an internal hom into an object dd, where the unit is the usual double-dual embedding δ c:c[[c,d],d]\delta_c: c \to [[c, d], d].

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