Adjoint functors

In category theory, a pair of adjoint functors between two categories C and D consists of two functors F : C → D and G : D → C and a natural equivalence consisting of bijective functions

φ_X,Y: Mor_D(F(X), Y) → Mor_C(X, G(Y))

for all objects X in C and Y in D. All such pairs of adjoint functors arise from universal constructions.

We say that F is left-adjoint to G and G is right-adjoint to F.

Every adjoint pair of functors defines a unit η, a natural transformation from Id_C to GF consisting of morphisms

η_X : X -> GF(X)

for every X in C. η_X is defined as φ_X,F(X) (id_F(X)). Analogously, one may define a co-unit ε, a natural transformation consisting of morphisms

ε_Y : FG(Y) → Y.

for every Y in D.

Examples

Free objects. If F : Set → Group is the functor assiging to each set X the free group over X, and if G : Group → Set is the forgetful functor assigning to each group its underlying set, then the universal property of the free group shows that F is left-adjoint to G. The unit of this adjoint pair is the embedding of a set X into the free group over X.

Free rings, free abelian groups, free modules etc. follow the same pattern.

Products. Let F : Group → Group² be the functor which assigns to every group X the pair (X, X) in the product category Group², and G : Group² → Group the functor which assigs to each pair (Y₁, Y₂) the product group Y₁×Y₂. The universal property of the product group shows that G is right-adjoint to F. The co-unit gives the natural projections from the product to the factors.

The cartesian product of sets, the product of rings, the product of topological spaces etc. follow the same pattern; it can also be extended in a straightforward manner to more than just two factors.

Coproducts. If F : Ab² → Ab assigns to every pair (X₁, X₂) of abelian groups their direct sum and if G : Ab → Ab² is the functor which assigns to every abelian group Y the pair (Y, Y), then F is left adjoint to G, again a consequence of the universal property of direct sums. The unit of the adjoint pair provides the natural embeddings from the factors into the direct sum.

Analogous examples are given by the direct sum of vector spaces and modules, by the free product[?] of groups and by the disjoint union of sets.

Kernels. Consider the category D of homomorphisms of abelian groups. If f₁ : A₁ → B₁ and f₂ : A₂ → B₂ are two objects of D, then a morphism from f₁ to f₂ is a pair (g_A, g_B) of morphisms such that g_Bf₁ = f₂g_A. Let G : D → Ab be the functor which assigns to each homomorphism its kernel and let F : Ab → D be the morphism which maps the group A to the homomorphism A → 0. Then G is right adjoint to F, which expresses the universal property of kernels, and the co-unit of this adjunction yields the natural embedding of a homomorphism's kernel into the homomorphism's domain.

A suitable variation of this example also shows that the kernel functors for vector spaces and for modules are right adjoints. Analogously, one can show that the cokernel functors for abelian groups, vector spaces and modules are left adjoints.

Ring extensions. Suppose R and S are rings, and ρ : R → S is a ring homomorphism. Then S can be seen as a (left) R-module, and the tensor product with S yields a functor F : R-Mod → S-Mod. Then F is left adjoint to the forgetful functor G : S-Mod → R-Mod.

Tensor products. If R is a ring and M is a right R module, then the tensor product with M yields a functor F : R-Mod → Ab. The functor G : Ab → R-Mod, defined by G(A) = Hom_Z(A, M) for every abelian group A, is a right adjoint to F.

Stone-Čech compactification. Let D be the category of compact Hausdorff spaces and G : D → Top be the forgetful functor which treats every compact Hausdorff space as a topological space. Then G has a left adjoint F : Top → D, the Stone-Čech compactification. The unit of this adjoint pair yields a continuous map from every topological space X into its Stone-Čech compactification. This map is an embedding (i.e. injective, continuous and open) if and only if X is a Tychonoff space.

Galois connections. Every partially ordered set can be viewed as a category (with a single morphism between x and y if and only if x ≤ y). A pair of adjoint functors between two partially ordered sets is called a Galois connection. See that article for a number of examples.

Facts about adjoints

If the functor F : C → D had two right-adjoints G₁ and G₂, then G₁ and G₂ are naturally isomorphic. The same is true for left-adjoints.

The most important property of adjoints is their continuity: every functor that has a left adjoint (and therefore is a right adjoint) is continuous; every functor that has a right adjoint (and therefore is a left adjoint) is cocontinuous (see limit (category theory)).

Not every functor G : D → C admits a left adjoint. If D is complete (see limit (category theory)), then the functors with left adjoints can be characterized by the Freyd Adjoint Functor Theorem: G has a left adjoint if and only if it is continuous and for every object x of C there exists a family of morphisms f_i : x → G(d_i) (where the indices i come from a set (not class) I), such that every morphism h : x → G(d) can be written as h = G(t) o f_i for some i in I and some morphism t : d_i → d in D.