dual adjunction



2-Category theory


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.


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 (