Adjointness

Adjointness is a formal relationship between two functors that expresses a generalized duality. An adjoint pair consists of functors F: C → D and G: D → C together with a natural isomorphism Hom_D(FX, Y) ≅ Hom_C(X, GY) for all objects X in C and Y in D. When this holds, F is called left adjoint to G and G right adjoint to F. Equivalently, there are unit and counit natural transformations η: Id_C ⇒ G∘F and ε: F∘G ⇒ Id_D satisfying the triangular identities. Adjointness encodes a universal property and often identifies F and G as best approximations to solving certain problems in their respective categories.

Key properties: left adjoints preserve all colimits; right adjoints preserve all limits. If an adjoint exists, it is unique up to a unique isomorphism.

Examples: the free group functor F: Set → Grp is left adjoint to the forgetful functor U: Grp → Set; in linear algebra, the tensor–hom adjunction gives Hom(V⊗W, X) ≅ Hom(V, Hom(W, X)).

Adjoints in linear algebra: the adjoint T* of a linear operator T with respect to an inner product satisfies ⟨Tx, y⟩ = ⟨x, T*y⟩ for all x,y. If a basis is chosen, T* is represented by the conjugate transpose. Basic identities include (AB)* = B* A*, (T*)* = T, and T** = T. Self-adjoint, skew-adjoint, unitary, and normal operators reflect symmetry properties relative to the inner product.