abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
While a (left or right) adjoint to a functor may be understood as the best approximation (from one side or the other) of a possibly non-existent inverse, any pair of adjoint functors restricts to an equivalence of categories on subcategories. These subcategories are sometimes known as the center of the adjunction, their objects are sometimes known as the fixed points of the adjunction.
The equivalences of categories that arise from fixed points of adjunctions this way are often known as dualities. Examples include Pontrjagin duality, Gelfand duality, Stone duality, and the Isbell duality between commutative rings and affine schemes (see Porst-Tholen 91).
(fixed point equivalence of an adjunction)
Let
be a pair of adjoint functors. Say that
an object $c \in \mathcal{C}$ is a fixed point of the adjunction if its adjunction unit is an isomorphism
and write
for the full subcategory on these fixed objects;
an object $d \in \mathcal{D}$ is a fixed point of the adjunction if its adjunction counit is an isomorphism
and write
for the full subcategory on these fixed objects.
Then the adjunction (co-)restrics to an adjoint equivalence on these full subcategories of fixed points:
It is sufficient to see that the functors (co-)restrict as claimed, for then the restricted adjunction unit/counit are isomorphisms by definition, and hence exhibit an adjoint equivalence.
Hence we need to show that
for $c \in \mathcal{C}_{fix} \hookrightarrow \mathcal{C}$ we have that $\epsilon_{L(c)}$ is an isomorphism;
for $d \in \mathcal{D}_{fix} \hookrightarrow \mathcal{D}$ we have that $\eta_{R(d)}$ is an isomorphism.
For the first case we claim that $L(\eta_{c})$ provides an inverse: by the triangle identity (?) it is a right inverse, but by assumption it is itself an invertible morphism, which implies that $\epsilon_{L(c)}$ is an isomorphism.
The second claim is formally dual.
Gelfand duality is (a further restriction of) the fixed point equivalence of the adjunction between compactly generated Hausdorff spaces and topological algebras over the complex numbers, given by using the complex numbers as dualizing object (Porst-Tholen 91, section 4-c).