Adjoint functors

In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e., constructions of objects having a certain universal property), such as the construction of a free group on a set in algebra, or the construction of the Stone-Čech compactification of a topological space in topology.

By definition, an adjunction between categories \(\mathcal{C}\) and \(\mathcal{D}\) is a pair of functors (assumed to be covariant)

\[\begin{aligned} F&: \mathcal{D} \rightarrow \mathcal{C}\\ G&: \mathcal{C} \rightarrow \mathcal{D} \end{aligned}\]

and, for all objects \(c\) in \(\mathcal{C}\) and \(d\) in \(\mathcal{D}\), a bijection between the respective morphism sets

\[\mathrm{hom}_{\mathcal{C}}(Fd,c) \cong \mathrm{hom}_{\mathcal{D}}(d,Gc)\]

such that this family of bijections is natural in \(c\) and \(d\). For locally small categories, naturality here means that there are natural isomorphisms between the pair of functors \(\mathcal{C}(F-,c) : \mathcal{D} \to \mathrm{Set^{\text{op}}}\) and \(\mathcal{D}(-,Gc) : \mathcal{D} \to \mathrm{Set^{\text{op}}}\) for a fixed \(c\) in \(\mathcal{C}\), and also the pair of functors \(\mathcal{C}(Fd,-) : \mathcal{C} \to \mathrm{Set}\) and \(\mathcal{D}(d,G-) : \mathcal{C} \to \mathrm{Set}\) for a fixed \(d\) in \(\mathcal{D}\). For other categories, naturality is defined as a generalisation of this.

The functor \(F\) is called a left adjoint functor or left adjoint to \(G\), while \(G\) is called a right adjoint functor or right adjoint to \(F\). We write \(F\dashv G\).

Condensed: the full section is in Wikipedia.

The terms adjoint and adjunct are both used, and are cognates: one is taken directly from Latin, the other from Latin via French. In the classic text Categories for the Working Mathematician, Mac Lane makes a distinction between the two. Given a family

\[\varphi_{cd}: \mathrm{hom}_{\mathcal{C}}(Fd,c) \cong \mathrm{hom}_{\mathcal{D}}(d,Gc)\]

of hom-set bijections, we call \(\varphi\) an adjunction or an adjunction between \(F\) and \(G\). If \(f\) is an arrow in \(\mathrm{hom}_{\mathcal{C}}(Fd,c)\), Mac Lane calls \(\varphi f\) the right adjunct of \(f\). The functor \(F\) is left adjoint to \(G\), and \(G\) is right adjoint to \(F\). (Note that \(G\) may have itself a right adjoint that is quite different from \(F\); see below for an example.)

In general, the phrases "\(F\) is a left adjoint" and "\(F\) has a right adjoint" are equivalent. We call \(F\) a left adjoint because it is applied to the left argument of \(\mathrm{hom}_{\mathcal{C}}\), and \(G\) a right adjoint because it is applied to the right argument of \(\mathrm{hom}_{\mathcal{D}}\).

If F is left adjoint to G, we also write \[F\dashv G.\]

The terminology comes from the Hilbert space idea of adjoint operators \(T\), \(U\) with \(\langle Ty,x\rangle = \langle y,Ux\rangle\), which is formally similar to the above relation between hom-sets. The analogy to adjoint maps of Hilbert spaces can be made precise in certain contexts.

, Saunders Mac Lane, Categories for the Working Mathematician

Common mathematical constructions are very often adjoint functors. Consequently, general theorems about left/right adjoint functors encode the details of many useful and otherwise non-trivial results. Such general theorems include the equivalence of the various definitions of adjoint functors, the uniqueness of a right adjoint for a given left adjoint, the fact that left/right adjoint functors respectively preserve colimits/limits (which are also found in every area of mathematics), and the general adjoint functor theorems giving conditions under which a given functor is a left/right adjoint.

In a sense, an adjoint functor is a way of giving the most efficient solution to some problem via a method that is formulaic. For example, an elementary problem in ring theory is how to turn a rng (which is like a ring that might not have a multiplicative identity) into a ring. The most efficient way is to adjoin an element '1' to the rng, adjoin all (and only) the elements that are necessary for satisfying the ring axioms (e.g. r+1 for each r in the ring), and impose no relations in the newly formed ring that are not forced by axioms. Moreover, this construction is formulaic in the sense that it works in essentially the same way for any rng.

This is rather vague, though suggestive, and can be made precise in the language of category theory: a construction is most efficient if it satisfies a universal property, and is formulaic if it defines a functor. Universal properties come in two types: initial properties and terminal properties. Since these are dual notions, it is only necessary to discuss one of them.

The idea of using an initial property is to set up the problem in terms of some auxiliary category E, so that the problem at hand corresponds to finding an initial object of E. This has an advantage that the optimization, the sense that the process finds the most efficient solution, means something rigorous and recognisable, rather like the attainment of a supremum. The category E is also formulaic in this construction, since it is always the category of elements of the functor to which one is constructing an adjoint.

Back to our example: take the given rng R, and make a category E whose objects are rng homomorphisms R → S, with S a ring having a multiplicative identity. The morphisms in E between R → S₁ and R → S₂ are commutative triangles of the form (R → S₁, R → S₂, S₁ → S₂) where S₁ → S₂ is a ring map (which preserves the identity). (Note that this is precisely the definition of the comma category of R over the inclusion of unitary rings into rng.) The existence of a morphism between R → S₁ and R → S₂ implies that S₁ is at least as efficient a solution as S₂ to our problem: S₂ can have more adjoined elements and/or more relations not imposed by axioms than S₁. Therefore, the assertion that an object R → R is initial in E, that is, that there is a morphism from it to any other element of E, means that the ring R* is a most efficient solution to our problem.

The two facts that this method of turning rngs into rings is most efficient and formulaic can be expressed simultaneously by saying that it defines an adjoint functor. More explicitly: Let F denote the above process of adjoining an identity to a rng, so F(R)=R. Let G denote the process of "forgetting" whether a ring S has an identity and considering it simply as a rng, so essentially G(S)=S. Then F is the left adjoint functor of G.

Note however that we haven't actually constructed R yet; it is an important and not altogether trivial algebraic fact that such a left adjoint functor R → R actually exists.

It is also possible to start with the functor F, and pose the following (vague) question: is there a problem to which F is the most efficient solution?

The notion that F is the most efficient solution to the problem posed by G is, in a certain rigorous sense, equivalent to the notion that G poses the most difficult problem that F solves.

This gives the intuition behind the fact that adjoint functors occur in pairs: if F is left adjoint to G, then G is right adjoint to F.

There are various equivalent definitions for adjoint functors:

• The definitions via universal morphisms are easy to state, and require minimal verifications when constructing an adjoint functor or proving two functors are adjoint. They are also the most analogous to our intuition involving optimizations.
• The definition via hom-sets makes symmetry the most apparent, and is the reason for using the word adjoint.
• The definition via counit, unit adjunction is convenient for proofs about functors that are known to be adjoint, because they provide formulas that can be directly manipulated.

The equivalency of these definitions is quite useful. Adjoint functors arise everywhere, in all areas of mathematics. Since the structure in any of these definitions gives rise to the structures in the others, switching between them makes implicit use of many details that would otherwise have to be repeated separately in every subject area.

The theory of adjoints has the terms left and right at its foundation, and there are many components that live in one of two categories C and D that are under consideration. Therefore, it can be helpful to choose letters in alphabetical order according to whether they live in the "lefthand" category C or the "righthand" category D, and also to write them down in this order whenever possible.

In this article for example, the letters X, F, f, ε will consistently denote things that live in the category C, the letters Y, G, g, η will consistently denote things that live in the category D, and whenever possible such things will be referred to in order from left to right (a functor F : D → C can be thought of as "living" where its outputs are, in C). If the arrows for the left adjoint functor F were drawn they would be pointing to the left; if the arrows for the right adjoint functor G were drawn they would be pointing to the right.

By definition, a functor \(F: \mathcal{D} \to \mathcal{C}\) is a left adjoint functor if for each object \(X\) in \(\mathcal{C}\) there exists a universal morphism from \(F\) to \(X\). Spelled out, this means that for each object \(X\) in \(\mathcal{C}\) there exists an object \(G(X)\) in \(\mathcal{D}\) and a morphism \(\varepsilon_X: F(G(X)) \to X\) such that for every object \(Y\) in \(\mathcal{D}\) and every morphism \(f: F(Y) \to X\) there exists a unique morphism \(g: Y \to G(X)\) with \(\varepsilon_X \circ F(g) = f\).

The latter equation is expressed by the following commutative diagram:

In this situation, one can show that \(G\) can be turned into a functor \(G :\mathcal{C} \to \mathcal{D}\) in a unique way such that \(\varepsilon_X \circ F(G(f)) = f \circ \varepsilon_{X'}\) for all morphisms \(f: X' \to X\) in \(\mathcal{C}\); \(F\) is then called a left adjoint to \(G\).

Similarly, we may define right-adjoint functors. A functor \(G: \mathcal{C} \to \mathcal{D}\) is a right adjoint functor if for each object \(Y\) in \(\mathcal{D}\), there exists a universal morphism from \(Y\) to \(G\). Spelled out, this means that for each object \(Y\) in \(\mathcal{D}\), there exists an object \(F(Y)\) in \(C\) and a morphism \(\eta_Y: Y \to G(F(Y))\) such that for every object \(X\) in \(\mathcal{C}\) and every morphism \(g: Y \to G(X)\) there exists a unique morphism \(f: F(Y) \to X\) with \(G(f) \circ \eta_Y = g\).

Again, this \(F\) can be uniquely turned into a functor \(F: \mathcal{D} \to \mathcal{C}\) such that \(G(F(g)) \circ \eta_Y = \eta_{Y'} \circ g\) for \(g: Y \to Y'\) a morphism in \(\mathcal{D}\); \(G\) is then called a right adjoint to \(F\).

It is true, as the terminology implies, that \(F\) is left adjoint to \(G\) if and only if \(G\) is right adjoint to \(F\).

These definitions via universal morphisms are often useful for establishing that a given functor is left or right adjoint, because they are minimalistic in their requirements. They are also intuitively meaningful in that finding a universal morphism is like solving an optimization problem.

Using hom-sets, an adjunction between two categories \(\mathcal{C}\) and \(\mathcal{D}\) can be defined as consisting of two functors \(F : \mathcal{D} \to \mathcal{C}\) and \(G : \mathcal{C} \to \mathcal{D}\) and a natural isomorphism \[\Phi:\mathrm{Hom}_\mathcal{C}(F-,-) \to \mathrm{Hom}_\mathcal{D}(-,G-).\] This specifies a family of bijections \[\Phi_{Y,X}:\mathrm{Hom}_\mathcal{C}(FY,X) \to \mathrm{Hom}_\mathcal{D}(Y,GX)\] for all objects \(X\in\mathcal{C}\) and \(Y\in\mathcal{D}.\)

In this situation, \(F\) is left adjoint to \(G\) and \(G\) is right adjoint to \(F\).

This definition is a logical compromise in that it is more difficult to establish its satisfaction than the universal morphism definitions, and has fewer immediate implications than the counit, unit definition. It is useful because of its obvious symmetry, and as a stepping-stone between the other definitions.

In order to interpret \(\Phi\) as a natural isomorphism, one must recognize \(\text{Hom}_\mathcal{C}(F-,-)\) and \(\text{Hom}_\mathcal{D}(-,G-)\) as functors. In fact, they are both bifunctors from \(\mathcal{D}^\text{op} \times \mathcal{C}\) to \(\mathbf{Set}\) (the category of sets). For details, see the article on hom-functors. Spelled out, the naturality of \(\Phi\) means that for all morphisms \(f : X \to X'\) in \(\mathcal{C}\) and all morphisms \(g : Y' \to Y\) in \(\mathcal{D}\) the following diagram commutes:

The vertical arrows in this diagram (\(\text{Hom}(g,Gf)\) and \(\text{Hom}(Fg,f)\)) are those induced by composition. Formally, \(\text{Hom}(Fg, f) : \text{Hom}_\mathcal{C}(FY, X) \to \text{Hom}_\mathcal{C}(FY', X')\) is given by \(h \mapsto f \circ h \circ Fg\) for each \(h \in \text{Hom}_\mathcal{C}(FY, X).\) \(\text{Hom}(g, Gf)\) is similar.

A third way of defining an adjunction between two categories \(\mathcal{C}\) and \(\mathcal{D}\) consists of two functors \(F : \mathcal{D} \to \mathcal{C}\) and \(G : \mathcal{C} \to \mathcal{D}\) and two natural transformations \[\begin{aligned} \varepsilon &: FG \to 1_\mathcal{C} \\ \eta &: 1_\mathcal{D} \to GF\end{aligned}\] respectively called the counit and the unit of the adjunction (terminology from universal algebra), such that the compositions \[F\xrightarrow\overset{}{\;F\eta\;}FGF\xrightarrow\overset{}{\;\varepsilon F\,}F\] \[G\xrightarrow\overset{}{\;\eta G\;}GFG\xrightarrow\overset{}{\;G \varepsilon\,}G\] are the identity morphisms \(1_F\) and \(1_G\) on F and G respectively.

In this situation we say that F is left adjoint to G and G is right adjoint to F, and may indicate this relationship by writing \((\varepsilon,\eta):F\dashv G\) , or, simply \(F\dashv G\) .

In equational form, the above conditions on \((\varepsilon,\eta)\) are the counit, unit equations \[\begin{aligned} 1_F &= \varepsilon F\circ F\eta\\ 1_G &= G\varepsilon \circ \eta G \end{aligned}\] which imply that for each \(X \in \mathcal{C}\) and each \(Y \in \mathcal{D},\) \[\begin{aligned} 1_{FY} &= \varepsilon_{FY}\circ F(\eta_Y) \\ 1_{GX} &= G(\varepsilon_X)\circ\eta_{GX}. \end{aligned}\]

Note that \(1_{\mathcal C}\) denotes the identify functor on the category \(\mathcal C\), \(1_F\) denotes the identity natural transformation from the functor F to itself, and \(1_{FY}\) denotes the identity morphism of the object \(FY\).

These equations are useful in reducing proofs about adjoint functors to algebraic manipulations. They are sometimes called the triangle identities, or sometimes the zig-zag equations because of the appearance of the corresponding string diagrams. A way to remember them is to first write down the nonsensical equation \(1=\varepsilon\circ\eta\) and then fill in either F or G in one of the two simple ways that make the compositions defined.

Note: The use of the prefix "co" in counit here is not consistent with the terminology of limits and colimits, because a colimit satisfies an initial property whereas the counit morphisms satisfy terminal properties, and dually for limit versus unit. The term unit here is borrowed from the theory of monads, where it looks like the insertion of the identity 1 into a monoid.

The idea of adjoint functors was introduced by Daniel Kan in 1958. Like many of the concepts in category theory, it was suggested by the needs of homological algebra, which was at the time devoted to computations. Those faced with giving tidy, systematic presentations of the subject would have noticed relations such as

\[\text{Hom}(FX, Y) = \text{Hom}(X, GY)\]

in the category of abelian groups, where F was the functor \(- \otimes A\) (i.e. take the tensor product with A), and G was the functor Hom(A,–) (this is now known as the tensor-hom adjunction). The use of the equals sign is an abuse of notation; those two groups are not really identical but there is a way of identifying them that is natural. It can be seen to be natural on the basis, firstly, that these are two alternative descriptions of the bilinear mappings from X × A to Y. That is, however, something particular to the case of tensor product. In category theory the 'naturality' of the bijection is subsumed in the concept of a natural isomorphism.

The construction of free groups is a common and illuminating example.

Let F : Set → Grp be the functor assigning to each set Y the free group generated by the elements of Y, and let G : Grp → Set be the forgetful functor, which assigns to each group X its underlying set. Then F is left adjoint to G:

Condensed: the full section is in Wikipedia.

Free objects are all examples of a left adjoint to a forgetful functor, which assigns to an algebraic object its underlying set. These algebraic free functors have generally the same description as in the detailed description of the free group situation above.

Products, pullbacks, equalizers, and kernels are all examples of the categorical notion of a limit. Any limit functor is right adjoint to a corresponding diagonal functor (provided the category has the type of limits in question), and the counit of the adjunction provides the defining maps from the limit object (i.e. from the diagonal functor on the limit, in the functor category). Below are some specific examples.

• Products. Let Π : Grp → Grp be the functor that assigns to each pair (X₁, X₂) the product group X₁×X₂, and let Δ : Grp → Grp be the diagonal functor that assigns to every group X the pair (X, X) in the product category Grp. The universal property of the product group shows that Π is right-adjoint to Δ. The counit of this adjunction is the defining pair of projection maps from X₁×X₂ to X₁ and X₂ which define the limit, and the unit is the diagonal inclusion of a group X into X×X (mapping x to (x,x)).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. More generally, any type of limit is right adjoint to a diagonal functor.
• 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 functor which maps the group A to the homomorphism A → 0. Then G is right adjoint to F, which expresses the universal property of kernels. The counit of this adjunction is the defining embedding of a homomorphism's kernel into the homomorphism's domain, and the unit is the morphism identifying a group A with the kernel of the homomorphism A → 0.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.