 Na matemática, specifically category theory, adjunction is a relationship that two functors may have, 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 (ou seja, 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.

Por definição, an adjunction between categories {estilo de exibição {matemática {C}}} e {estilo de exibição {matemática {D}}} is a pair of functors (assumed to be covariant) {estilo de exibição F:{matemática {D}}rightarrow {matemática {C}}} e {estilo de exibição G:{matemática {C}}rightarrow {matemática {D}}} e, for all objects {estilo de exibição X} dentro {estilo de exibição {matemática {C}}} e {estilo de exibição Y} dentro {estilo de exibição {matemática {D}}} a bijection between the respective morphism sets {matemática de estilo de exibição {hom} _{matemática {C}}(FY,X)cong mathrm {hom} _{matemática {D}}(S,GX)} such that this family of bijections is natural in {estilo de exibição X} e {estilo de exibição Y} . Naturality here means that there are natural isomorphisms between the pair of functors {estilo de exibição {matemática {C}}(F-,X):{matemática {D}}para matemática {Definir} } e {estilo de exibição {matemática {D}}(-,GX):{matemática {D}}para matemática {Definir} } for a fixed {estilo de exibição X} dentro {estilo de exibição {matemática {C}}} , and also the pair of functors {estilo de exibição {matemática {C}}(FY,-):{matemática {C}}para matemática {Definir} } e {estilo de exibição {matemática {D}}(S,G-):{matemática {C}}para matemática {Definir} } for a fixed {estilo de exibição Y} dentro {estilo de exibição {matemática {D}}} .

The functor {estilo de exibição F} is called a left adjoint functor or left adjoint to {estilo de exibição G} , enquanto {estilo de exibição G} is called a right adjoint functor or right adjoint to {estilo de exibição F} .

An adjunction between categories {estilo de exibição {matemática {C}}} e {estilo de exibição {matemática {D}}} is somewhat akin to a "weak form" of an equivalence between {estilo de exibição {matemática {C}}} e {estilo de exibição {matemática {D}}} , and indeed every equivalence is an adjunction. In many situations, an adjunction can be "upgraded" to an equivalence, by a suitable natural modification of the involved categories and functors.

If F is left adjoint to G, we also write {displaystyle Fdashv G.} The terminology comes from the Hilbert space idea of adjoint operators {estilo de exibição T} , {estilo de exibição U} com {displaystyle langle Ty,xrangle =langle y,Uxrangle } , 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. Introduction and Motivation The slogan is "Adjoint functors arise everywhere".

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 which is formulaic. Por exemplo, 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 which are necessary for satisfying the ring axioms (por exemplo. r+1 for each r in the ring), and impose no relations in the newly formed ring that are not forced by axioms. Além disso, 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 is 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. Portanto, the assertion that an object R → R* is initial in E, isso é, 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. Mais explicitamente: 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.

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.

Formal definitions 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 which 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 a great deal of tedious 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 which live in one of two categories C and D which 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 which live in the category C, the letters Y, G, g, η will consistently denote things which 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, em C).

Definition via universal morphisms By definition, a functor {estilo de exibição F:Dto C} is a left adjoint functor if for each object {estilo de exibição X} dentro {estilo de exibição C} there exists a universal morphism from {estilo de exibição F} para {estilo de exibição X} . Spelled out, this means that for each object {estilo de exibição X} dentro {estilo de exibição C} there exists an object {estilo de exibição G(X)} dentro {estilo de exibição D} and a morphism {displaystyle epsilon _{X}:F(G(X))para X} such that for every object {estilo de exibição Y} dentro {estilo de exibição D} and every morphism {estilo de exibição f:F(S)para X} there exists a unique morphism {estilo de exibição g:Yto G(X)} com {displaystyle epsilon _{X}circ F(g)=f} .

The latter equation is expressed by the following commutative diagram: In this situation, one can show that {estilo de exibição G} can be turned into a functor {estilo de exibição G:Cto D} in a unique way such that {displaystyle epsilon _{X}circ F(G(f))=fcirc epsilon _{X'}} for all morphisms {estilo de exibição f:X'to X} dentro {estilo de exibição C} ; {estilo de exibição F} is then called a left adjoint to {estilo de exibição G} .

De forma similar, we may define right-adjoint functors. A functor {estilo de exibição G:Cto D} is a right adjoint functor if for each object {estilo de exibição Y} dentro {estilo de exibição D} , there exists a universal morphism from {estilo de exibição Y} para {estilo de exibição G} . Spelled out, this means that for each object {estilo de exibição Y} dentro {estilo de exibição D} , there exists an object {estilo de exibição F(S)} dentro {estilo de exibição C} and a morphism {estilo de exibição eta _{S}:Yto G(F(S))} such that for every object {estilo de exibição X} dentro {estilo de exibição C} and every morphism {estilo de exibição g:Yto G(X)} there exists a unique morphism {estilo de exibição f:F(S)para X} com {estilo de exibição G(f)circ eta _{S}=g} .

Novamente, this {estilo de exibição F} can be uniquely turned into a functor {estilo de exibição F:Dto C} de tal modo que {estilo de exibição G(F(g))circ eta _{S}=eta _{Y'}circ g} por {estilo de exibição g:Yto Y'} a morphism in {estilo de exibição D} ; {estilo de exibição G} is then called a right adjoint to {estilo de exibição F} .

It is true, as the terminology implies, este {estilo de exibição F} is left adjoint to {estilo de exibição G} se e apenas se {estilo de exibição G} is right adjoint to {estilo de exibição 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.

Definition via Hom-set adjunction A hom-set adjunction between two categories C and D consists of two functors F : D → C and G : C → D and a natural isomorphism {estilo de exibição Phi :matemática {hom} _{C}(F-,-)para matemática {hom} _{D}(-,G-)} .

This specifies a family of bijections {displaystyle Phi _{S,X}:matemática {hom} _{C}(FY,X)para matemática {hom} _{D}(S,GX)} for all objects X in C and Y in 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 somewhat more difficult to satisfy 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 Φ as a natural isomorphism, one must recognize homC(F–, –) and homD(–, G–) as functors. Na verdade, they are both bifunctors from Dop × C to Set (the category of sets). For details, see the article on hom functors. Explicitamente, the naturality of Φ means that for all morphisms f : X → X′ in C and all morphisms g : Y′ → Y in D the following diagram commutes: The vertical arrows in this diagram are those induced by composition. Formalmente, Hom(Fg, f) : HomC(FY, X) → HomC(FY′, X′) is given by h → f o h o Fg for each h in HomC(FY, X). Hom(g, Gf) É similar.

Definition via counit–unit adjunction A counit–unit adjunction between two categories C and D consists of two functors F : D → C and G : C → D and two natural transformations {estilo de exibição {começar{alinhado}varepsilon &:FGto 1_{matemática {C}}\eta &:1_{matemática {D}}to GFend{alinhado}}} respectively called the counit and the unit of the adjunction (terminology from universal algebra), such that the compositions {estilo de exibição F{seta para a direita {;Feta ;}}FGF{seta para a direita {;varepsilon F,}}F} {estilo de exibição G{seta para a direita {;eta G;}}GFG{seta para a direita {;Gvarepsilon ,}}G} are the identity transformations 1F and 1G 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 {estilo de exibição (varepsilon ,e ):Fdashv G} , or simply {displaystyle Fdashv G} .

In equation form, the above conditions on (e,o) are the counit–unit equations {estilo de exibição {começar{alinhado}1_{F}&=varepsilon Fcirc Feta \1_{G}&=Gvarepsilon circ eta Gend{alinhado}}} which mean that for each X in C and each Y in D, {estilo de exibição {começar{alinhado}1_{FY}&=varepsilon _{FY}circ F(e _{S})\1_{GX}&=G(varepsilon _{X})circ eta _{GX}fim{alinhado}}} .

Observe que {displaystyle 1_{matemática {C}}} denotes the identify functor on the category {estilo de exibição {matemática {C}}} , {displaystyle 1_{F}} denotes the identity natural transformation from the functor F to itself, e {displaystyle 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 {displaystyle 1=varepsilon circ eta } and then fill in either F or G in one of the two simple ways which make the compositions defined.

Observação: 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 will satisfy terminal properties, and dually. The term unit here is borrowed from the theory of monads where it looks like the insertion of the identity 1 into a monoid.

History 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), S) = hom(X, G(S)) in the category of abelian groups, where F was the functor {displaystyle -otimes A} (ou seja. take the tensor product with A), and G was the functor hom(UMA,–) (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. Aquilo é, Contudo, 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.

Ubiquity If one starts looking for these adjoint pairs of functors, they turn out to be very common in abstract algebra, and elsewhere as well. The example section below provides evidence of this; furthermore, universal constructions, which may be more familiar to some, give rise to numerous adjoint pairs of functors.

In accordance with the thinking of Saunders Mac Lane, any idea, such as adjoint functors, that occurs widely enough in mathematics should be studied for its own sake.[citação necessária] Concepts can be judged according to their use in solving problems, as well as for their use in building theories. The tension between these two motivations was especially great during the 1950s when category theory was initially developed. Enter Alexander Grothendieck, who used category theory to take compass bearings in other work—in functional analysis, homological algebra and finally algebraic geometry.

It is probably wrong to say that he promoted the adjoint functor concept in isolation: but recognition of the role of adjunction was inherent in Grothendieck's approach. Por exemplo, one of his major achievements was the formulation of Serre duality in relative form—loosely, in a continuous family of algebraic varieties. The entire proof turned on the existence of a right adjoint to a certain functor. This is something undeniably abstract, and non-constructive[discuss], but also powerful in its own way.

Examples 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: Initial morphisms. For each set Y, the set GFY is just the underlying set of the free group FY generated by Y. Deixar {estilo de exibição eta _{S}:Yto GFY} be the set map given by "inclusion of generators". This is an initial morphism from Y to G, because any set map from Y to the underlying set GW of some group W will factor through {estilo de exibição eta _{S}:Yto GFY} via a unique group homomorphism from FY to W. This is precisely the universal property of the free group on Y.

Terminal morphisms. For each group X, the group FGX is the free group generated freely by GX, the elements of X. Deixar {displaystyle varepsilon _{X}:FGXto X} be the group homomorphism which sends the generators of FGX to the elements of X they correspond to, which exists by the universal property of free groups. Then each {estilo de exibição (GX,varepsilon _{X})} is a terminal morphism from F to X, because any group homomorphism from a free group FZ to X will factor through {displaystyle varepsilon _{X}:FGXto X} via a unique set map from Z to GX. Isso significa que (F,G) is an adjoint pair.

Hom-set adjunction. Group homomorphisms from the free group FY to a group X correspond precisely to maps from the set Y to the set GX: each homomorphism from FY to X is fully determined by its action on generators, another restatement of the universal property of free groups. One can verify directly that this correspondence is a natural transformation, which means it is a hom-set adjunction for the pair (F,G).

counit–unit adjunction. One can also verify directly that ε and η are natural. Então, a direct verification that they form a counit–unit adjunction {estilo de exibição (varepsilon ,e ):Fdashv G} é o seguinte: The first counit–unit equation {displaystyle 1_{F}=varepsilon Fcirc Feta } says that for each set Y the composition {displaystyle FY{seta para a direita {;F(e _{S});}}FGFY{seta para a direita {;varepsilon _{FY},}}FY} should be the identity. The intermediate group FGFY is the free group generated freely by the words of the free group FY. (Think of these words as placed in parentheses to indicate that they are independent generators.) The arrow {estilo de exibição F(e _{S})} is the group homomorphism from FY into FGFY sending each generator y of FY to the corresponding word of length one (y) as a generator of FGFY. The arrow {displaystyle varepsilon _{FY}} is the group homomorphism from FGFY to FY sending each generator to the word of FY it corresponds to (so this map is "dropping parentheses"). The composition of these maps is indeed the identity on FY.

The second counit–unit equation {displaystyle 1_{G}=Gvarepsilon circ eta G} says that for each group X the composition {displaystyle GX{seta para a direita {;e _{GX};}}GFGX{seta para a direita {;G(varepsilon _{X}),}}GX} should be the identity. The intermediate set GFGX is just the underlying set of FGX. The arrow {estilo de exibição eta _{GX}} é o "inclusion of generators" set map from the set GX to the set GFGX. The arrow {estilo de exibição G(varepsilon _{X})} is the set map from GFGX to GX which underlies the group homomorphism sending each generator of FGX to the element of X it corresponds to ("dropping parentheses"). The composition of these maps is indeed the identity on GX.

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, fibred products, 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 (ou seja. from the diagonal functor on the limit, in the functor category). Below are some specific examples.

Products Let Π : Grp2 → Grp the functor which assigns to each pair (X1, X2) the product group X1×X2, and let Δ : Grp → Grp2 be the diagonal functor which assigns to every group X the pair (X, X) in the product category Grp2. 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. De forma geral, 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 (gA, gB) of morphisms such that gBf1 = f2gA. 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. Colimits and diagonal functors Coproducts, fibred coproducts, 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.

As is the case for Galois groups, the real interest lies often in refining a correspondence to a duality (ou seja. antitone order isomorphism). A treatment of Galois theory along these lines by Kaplansky was influential in the recognition of the general structure here.

Define a category based on {estilo de exibição mathbb {R} } , with objects being the real numbers, and the morphisms being "affine functions evaluated at a point". Aquilo é, for any affine function {estilo de exibição f(x)=ax+b} and any real number {estilo de exibição r} , define a morphism {estilo de exibição (r,f):rto f(r)} .

Define a category based on {estilo de exibição M(mathbb {R} )} , the set of probability distribution on {estilo de exibição mathbb {R} } with finite expectation. Define morphisms on {estilo de exibição M(mathbb {R} )} Como "affine functions evaluated at a distribution". Aquilo é, for any affine function {estilo de exibição f(x)=ax+b} and any {displaystyle mu in M(mathbb {R} )} , define a morphism {estilo de exibição (dentro ,f):rto mu circ f^{-1}} .

Então, the Dirac delta measure defines a functor: {delta de estilo de exibição :xmapsto delta _{x}} , and the expectation defines another functor {estilo de exibição mathbb {E} :mu mapsto mathbb {E} [dentro ]} , and they are adjoint: {estilo de exibição mathbb {E} dashv delta } . (Somewhat disconcertingly, {estilo de exibição mathbb {E} } is the left adjoint, even though {estilo de exibição mathbb {E} } é "forgetful" e {delta de estilo de exibição } é "gratuitamente".) Adjunctions in full 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 A functor F : D → C called the left adjoint A functor G : C → D called the right adjoint A natural isomorphism Φ : homC(F–,–) → homD(–,G–) A natural transformation ε : FG → 1C called the counit A natural transformation η : 1D → GF called the unit An equivalent formulation, where X denotes any object of C and Y denotes any object of D, é o seguinte: For every C-morphism f : FY → X, there is a unique D-morphism ΦY, X(f) = g : Y → GX such that the diagrams below commute, and for every D-morphism g : Y → GX, there is a unique C-morphism Φ−1Y, X(g) = f : FY → X in C such that the diagrams below commute: From this assertion, one can recover that: The transformations ε, o, and Φ are related by the equations {estilo de exibição {começar{alinhado}f=Phi _{S,X}^{-1}(g)&=varepsilon _{X}circ F(g)&in &,,matemática {hom} _{C}(F(S),X)\g=Phi _{S,X}(f)&=G(f)circ eta _{S}&in &,,matemática {hom} _{D}(S,G(X))\Phi _{GX,X}^{-1}(1_{GX})&=varepsilon _{X}&in &,,matemática {hom} _{C}(FG(X),X)\Phi _{S,FY}(1_{FY})&=eta _{S}&in &,,matemática {hom} _{D}(S,GF(S))\fim{alinhado}}} The transformations ε, η satisfy the counit–unit equations {estilo de exibição {começar{alinhado}1_{FY}&=varepsilon _{FY}circ F(e _{S})\1_{GX}&=G(varepsilon _{X})circ eta _{GX}fim{alinhado}}} Each pair (GX, εX) is a terminal morphism from F to X in C Each pair (FY, ηY) is an initial morphism from Y to G in D In particular, the equations above allow one to define Φ, e, and η in terms of any one of the three. No entanto, 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.

Construct a functor F : D → C and a natural transformation η. For each object Y in D, choose an initial morphism (F(S), ηY) from Y to G, so that ηY : Y → G(F(S)). We have the map of F on objects and the family of morphisms η. For each f : Y0 → Y1, Como (F(Y0), ηY0) is an initial morphism, then factorize ηY1 o f with ηY0 and get F(f) : F(Y0) → F(Y1). This is the map of F on morphisms. The commuting diagram of that factorization implies the commuting diagram of natural transformations, so η : 1D → G o F is a natural transformation. Uniqueness of that factorization and that G is a functor implies that the map of F on morphisms preserves compositions and identities. Construct a natural isomorphism Φ : homC(F-,-) → homD(-,G-). For each object X in C, each object Y in D, Como (F(S), ηY) is an initial morphism, then ΦY, X is a bijection, where ΦY, X(f : F(S) → X) = G(f) o ηY. η is a natural transformation, G is a functor, then for any objects X0, X1 in C, any objects Y0, Y1 in D, any x : X0 → X1, any y : Y1 → Y0, we have ΦY1, X1(x o f o F(y)) = G(x) o G(f) o G(F(y)) o ηY1 = G(x) o G(f) o ηY0 o y = G(x) o ΦY0, X0(f) o y, and then Φ is natural in both arguments.

An analogous statement characterizes those functors with a right adjoint.

An important special case is that of locally presentable categories. Se {estilo de exibição F:Cto D} is a functor between locally presentable categories, then F has a right adjoint if and only if F preserves small colimits F has a left adjoint if and only if F preserves small limits and is an accessible functor 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.

Por outro lado, if F is left adjoint to G, and G is naturally isomorphic to G′ then F is also left adjoint to G′. De forma geral, if 〈F, G, e, η〉 is an adjunction (with counit–unit (e,o)) and σ : F → F′ τ : G → G′ are natural isomorphisms then 〈F′, G′, ε′, η′〉 is an adjunction where {estilo de exibição {começar{alinhado}eta '&=(tau ast sigma )circ eta \varepsilon '&=varepsilon circ (sigma^{-1}ast tau ^{-1}).fim{alinhado}}} Aqui {displaystyle circ } denotes vertical composition of natural transformations, e {displaystyle ast } denotes horizontal composition.

Composition Adjunctions can be composed in a natural fashion. Especificamente, if 〈F, G, e, η〉 is an adjunction between C and D and 〈F′, G′, ε′, η′〉 is an adjunction between D and E then the functor {displaystyle Fcirc F':Erightarrow C} is left adjoint to {displaystyle G'circ G:Cto E.} Mais precisamente, there is an adjunction between F F' and G' G with unit and counit given respectively by the compositions: {estilo de exibição {começar{alinhado}&1_{matemática {E}}{seta para a direita {e '}}G'F'{seta para a direita {G'eta F'}}G'GFF'\&FF'G'G{seta para a direita {Fvarepsilon 'G}}FG{seta para a direita {varepsilon }}1_{matemática {C}}.fim{alinhado}}} This new adjunction is called the composition of the two given adjunctions.

Since there is also a natural way to define an identity adjunction between a category C and itself, one can then form a category whose objects are all small categories and whose morphisms are 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) é contínuo (ou seja. commutes with limits in the category theoretical sense); every functor that has a right adjoint (and therefore is a left adjoint) is cocontinuous (ou seja. commutes with colimits).

Além disso, if both C and D are additive categories (ou seja. preadditive categories with all finite biproducts), then any pair of adjoint functors between them are automatically additive.

Relationships 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. Por outro lado, if there exists a universal morphism to a functor G : C → D from every object of D, then G has a left adjoint.

No entanto, 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 (equivalentemente, 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, ou seja. an adjunction whose unit and counit are isomorphisms.

Every adjunction 〈F, G, e, 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, então, adjoints are "generalizado" inverses. Note however that a right inverse of F (ou seja. 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, e, η〉 gives rise to an associated monad 〈T, o, μ〉 in the category D. The functor {estilo de exibição T:{matemática {D}}para {matemática {D}}} is given by T = GF. The unit of the monad {estilo de exibição eta :1_{matemática {D}}to T} is just the unit η of the adjunction and the multiplication transformation {mostre o estilo dele :T^{2}to T,} is given by μ = GεF. Duplamente, the triple 〈FG, e, 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.