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, 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.
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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 φ c d : h o m C ( F d , c ) ≅ h o m D ( d , G c ) {\displaystyle \varphi _{cd}:\mathrm {hom} _{\mathcal {C}}(Fd,c)\cong \mathrm {hom} _{\mathcal {D}}(d,Gc)} of hom-set bijections, we call φ {\displaystyle \varphi } an adjunction or an adjunction between F {\displaystyle F} and G {\displaystyle G} . If f {\displaystyle f} is an arrow in h o m C ( F d , c ) {\displaystyle \mathrm {hom} _{\mathcal {C}}(Fd,c)} , Mac Lane calls φ f {\displaystyle \varphi f} the right adjunct of f {\displaystyle f} . The functor F {\displaystyle F} is left adjoint to G {\displaystyle G} , and G {\displaystyle G} is right adjoint to F {\displaystyle F} . (Note that G {\displaystyle G} may have itself a right adjoint that is quite different from F {\displaystyle F} ; see below for an example.)
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The slogan is "Adjoint functors arise everywhere". — 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.
Solutions to optimization problems
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.
Symmetry of optimization problems
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 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.
Conventions
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.
Definition via universal morphisms
By definition, a functor F : D → C {\displaystyle F:{\mathcal {D}}\to {\mathcal {C}}} is a left adjoint functor if for each object X {\displaystyle X} in C {\displaystyle {\mathcal {C}}} there exists a universal morphism from F {\displaystyle F} to X {\displaystyle X} . Spelled out, this means that for each object X {\displaystyle X} in C {\displaystyle {\mathcal {C}}} there exists an object G ( X ) {\displaystyle G(X)} in D {\displaystyle {\mathcal {D}}} and a morphism ε X : F ( G ( X ) ) → X {\displaystyle \varepsilon _{X}:F(G(X))\to X} such that for every object Y {\displaystyle Y} in D {\displaystyle {\mathcal {D}}} and every morphism f : F ( Y ) → X {\displaystyle f:F(Y)\to X} there exists a unique morphism g : Y → G ( X ) {\displaystyle g:Y\to G(X)} with ε X ∘ F ( g ) = f {\displaystyle \varepsilon _{X}\circ F(g)=f} .
Definition via hom-sets
Using hom-sets, an adjunction between two categories C {\displaystyle {\mathcal {C}}} and D {\displaystyle {\mathcal {D}}} can be defined as consisting of two functors F : D → C {\displaystyle F:{\mathcal {D}}\to {\mathcal {C}}} and G : C → D {\displaystyle G:{\mathcal {C}}\to {\mathcal {D}}} and a natural isomorphism Φ : H o m C ( F − , − ) → H o m D ( − , G − ) . {\displaystyle \Phi :\mathrm {Hom} _{\mathcal {C}}(F-,-)\to \mathrm {Hom} _{\mathcal {D}}(-,G-).} This specifies a family of bijections Φ Y , X : H o m C ( F Y , X ) → H o m D ( Y , G X ) {\displaystyle \Phi _{Y,X}:\mathrm {Hom} _{\mathcal {C}}(FY,X)\to \mathrm {Hom} _{\mathcal {D}}(Y,GX)} for all objects X ∈ C {\displaystyle X\in {\mathcal {C}}} and Y ∈ D . {\displaystyle Y\in {\mathcal {D}}.}
Definition via counit–unit
A third way of defining an adjunction between two categories C {\displaystyle {\mathcal {C}}} and D {\displaystyle {\mathcal {D}}} consists of two functors F : D → C {\displaystyle F:{\mathcal {D}}\to {\mathcal {C}}} and G : C → D {\displaystyle G:{\mathcal {C}}\to {\mathcal {D}}} and two natural transformations ε : F G → 1 C η : 1 D → G F {\displaystyle {\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 → F η F G F → ε F F {\displaystyle F\xrightarrow {\overset {}{\;F\eta \;}} FGF\xrightarrow {\overset {}{\;\varepsilon F\,}} F} G → η G G F G → G ε G {\displaystyle G\xrightarrow {\overset {}{\;\eta G\;}} GFG\xrightarrow {\overset {}{\;G\varepsilon \,}} G} are the identity morphisms 1 F {\displaystyle 1_{F}} and 1 G {\displaystyle 1_{G}} on F and G respectively.
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 Hom ( F X , Y ) = Hom ( X , G Y ) {\displaystyle {\text{Hom}}(FX,Y)={\text{Hom}}(X,GY)} in the category of abelian groups, where F was the functor − ⊗ A {\displaystyle -\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.
Free groups
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:
Free constructions and forgetful functors
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.
Diagonal functors and limits
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.
Colimits and diagonal functors
Coproducts, pushouts, coequalizers, and cokernels are all examples of the categorical notion of a colimit. Any colimit functor is left adjoint to a corresponding diagonal functor (provided the category has the type of colimits in question), and the unit of the adjunction provides the defining maps into the colimit object. Below are some specific examples.
Further examples
Every partially ordered set can be viewed as a category (where the elements of the poset become the category's objects and we have a single morphism from x to y if and only if x ≤ y). A pair of adjoint functors between two partially ordered sets is called a Galois connection (or, if it is contravariant, an antitone Galois connection). See that article for a number of examples: the case of Galois theory of course is a leading one. Any Galois connection gives rise to closure operators and to inverse order-preserving bijections between the corresponding closed elements. As is the case for Galois groups, the real interest lies often in refining a correspondence to a duality (i.e. antitone order isomorphism). A treatment of Galois theory along these lines by Kaplansky was influential in the recognition of the general structure here.
There are hence numerous functors and natural transformations associated with every adjunction, and only a small portion is sufficient to determine the rest. An adjunction between categories C and D consists of An equivalent formulation, where X denotes any object of C and Y denotes any object of D, is as follows: From this assertion, one can recover that: In particular, the equations above allow one to define Φ, ε, and η in terms of any one of the three. However, the adjoint functors F and G alone are in general not sufficient to determine the adjunction. The equivalence of these situations is demonstrated below.
Universal morphisms induce hom-set adjunction
Given a right adjoint functor G : C → D, in the sense of initial morphisms, one may construct the induced hom-set adjunction by doing the following steps. A similar argument allows one to construct a hom-set adjunction from the terminal morphisms to a left adjoint functor. (The construction that starts with a right adjoint is slightly more common, since the right adjoint in many adjoint pairs is a trivially defined inclusion or forgetful functor.)
counit–unit adjunction induces hom-set adjunction
Given functors F : D → C, G : C → D, and a counit–unit adjunction (ε, η) : F ⊣ G, we can construct a hom-set adjunction by finding the natural transformation Φ : homC(F−,−) → homD(−,G−) in the following steps:
Hom-set adjunction induces all of the above
Given functors F : D → C, G : C → D, and a hom-set adjunction Φ : homC(F−,−) → homD(−,G−), one can construct a counit–unit adjunction ( ε , η ) : F ⊣ G , {\displaystyle (\varepsilon ,\eta ):F\dashv G,} which defines families of initial and terminal morphisms, in the following steps:
Existence
Not every functor G : C → D admits a left adjoint. If C is a complete category, then the functors with left adjoints can be characterized by the adjoint functor theorem of Peter J. Freyd: G has a left adjoint if and only if it is continuous and a certain smallness condition is satisfied: for every object Y of D there exists a family of morphisms where the indices i come from a set I, not a proper class, such that every morphism An analogous statement characterizes those functors with a right adjoint. An important special case is that of locally presentable categories. If F : C → D {\displaystyle F:C\to D} is a functor between locally presentable categories, then
Uniqueness
If the functor F : D → C has two right adjoints G and G′, then G and G′ are naturally isomorphic. The same is true for left adjoints. Conversely, if F is left adjoint to G, and G is naturally isomorphic to G′ then F is also left adjoint to G′. More generally, if ⟨F, G, ε, η⟩ is an adjunction (with counit–unit (ε,η)) and {{block indent|σ : F → F′ {{block indent|τ : G → G′ are natural isomorphisms then ⟨F′, G′, ε′, η′⟩ is an adjunction where η ′ = ( τ ∗ σ ) ∘ η ε ′ = ε ∘ ( σ − 1 ∗ τ − 1 ) . {\displaystyle {\begin{aligned}\eta '&=(\tau \ast \sigma )\circ \eta \\\varepsilon '&=\varepsilon \circ (\sigma ^{-1}\ast \tau ^{-1}).\end{aligned}}} Here ∘ {\displaystyle \circ } denotes vertical composition of natural transformations, and ∗ {\displaystyle \ast } denotes horizontal composition.
Composition
Adjunctions can be composed in a natural fashion. Specifically, if ⟨F, G, ε, η⟩ is an adjunction between C and D and ⟨F′, G′, ε′, η′⟩ is an adjunction between D and E then the functor F ∘ F ′ : E → C {\displaystyle F\circ F':E\rightarrow C} is left adjoint to G ′ ∘ G : C → E . {\displaystyle G'\circ G:C\to E.} More precisely, there is an adjunction between F F′ and G′ G with unit and counit given respectively by the compositions: 1 E → η ′ G ′ F ′ → G ′ η F ′ G ′ G F F ′ F F ′ G ′ G → F ε ′ G F G → ε 1 C . {\displaystyle {\begin{aligned}&1_{\mathcal {E}}{\xrightarrow {\eta '}}G'F'{\xrightarrow {G'\eta F'}}G'GFF'\\&FF'G'G{\xrightarrow {F\varepsilon 'G}}FG{\xrightarrow {\varepsilon }}1_{\mathcal {C}}.\end{aligned}}} This new adjunction is called the composition of the two given adjunctions.
Limit preservation
The most important property of adjoints is their continuity: every functor that has a left adjoint (and therefore is a right adjoint) is continuous (i.e. commutes with limits in the category theoretical sense); every functor that has a right adjoint (and therefore is a left adjoint) is cocontinuous (i.e. commutes with colimits). Since many common constructions in mathematics are limits or colimits, this provides a wealth of information. For example:
Additivity
If C and D are preadditive categories and F : D → C is an additive functor with a right adjoint G : C → D, then G is also an additive functor and the hom-set bijections Φ Y , X : h o m C ( F Y , X ) ≅ h o m D ( Y , G X ) {\displaystyle \Phi _{Y,X}:\mathrm {hom} _{\mathcal {C}}(FY,X)\cong \mathrm {hom} _{\mathcal {D}}(Y,GX)} are, in fact, isomorphisms of abelian groups. Dually, if G is additive with a left adjoint F, then F is also additive. Moreover, if both C and D are additive categories (i.e. preadditive categories with all finite biproducts), then any pair of adjoint functors between them are automatically additive.
Universal constructions
As stated earlier, an adjunction between categories C and D gives rise to a family of universal morphisms, one for each object in C and one for each object in D. Conversely, if there exists a universal morphism to a functor G : C → D from every object of D, then G has a left adjoint. However, universal constructions are more general than adjoint functors: a universal construction is like an optimization problem; it gives rise to an adjoint pair if and only if this problem has a solution for every object of D (equivalently, every object of C).
Equivalences of categories
If a functor F : D → C is one half of an equivalence of categories then it is the left adjoint in an adjoint equivalence of categories, i.e. an adjunction whose unit and counit are isomorphisms. Every adjunction ⟨F, G, ε, η⟩ extends an equivalence of certain subcategories. Define C1 as the full subcategory of C consisting of those objects X of C for which εX is an isomorphism, and define D1 as the full subcategory of D consisting of those objects Y of D for which ηY is an isomorphism. Then F and G can be restricted to D1 and C1 and yield inverse equivalences of these subcategories. In a sense, then, adjoints are "generalized" inverses. Note however that a right inverse of F (i.e. a functor G such that FG is naturally isomorphic to 1D) need not be a right (or left) adjoint of F. Adjoints generalize two-sided inverses.
Monads
Every adjunction ⟨F, G, ε, η⟩ gives rise to an associated monad ⟨T, η, μ⟩ in the category D. The functor T : D → D {\displaystyle T:{\mathcal {D}}\to {\mathcal {D}}} is given by T = GF. The unit of the monad η : 1 D → T {\displaystyle \eta :1_{\mathcal {D}}\to T} is just the unit η of the adjunction and the multiplication transformation μ : T 2 → T {\displaystyle \mu :T^{2}\to T\,} is given by μ = GεF. Dually, the triple ⟨FG, ε, FηG⟩ defines a comonad in C. Every monad arises from some adjunction—in fact, typically from many adjunctions—in the above fashion. Two constructions, called the category of Eilenberg–Moore algebras and the Kleisli category are two extremal solutions to the problem of constructing an adjunction that gives rise to a given monad.
