Editing Adjoint functors (section)

===Further examples===

==== Algebra ====
* '''Adjoining an identity to a [[Rng (algebra)|rng]].'''  This example was discussed in the motivation section above.  Given a rng ''R'', a multiplicative identity element can be added by taking ''R''x'''Z''' and defining a '''Z'''-bilinear product with (r,0)(0,1) = (0,1)(r,0) = (r,0), (r,0)(s,0) = (rs,0), (0,1)(0,1) = (0,1). This constructs a left adjoint to the functor taking a ring to the underlying rng.
* '''Adjoining an identity to a [[semigroup]].''' Similarly, given a semigroup ''S'', we can add an identity element and obtain a [[monoid]] by taking the [[disjoint union]] ''S'' <math>\sqcup</math> {1} and defining a binary operation on it such that it extends the operation on ''S'' and 1 is an identity element. This construction gives a functor that is  a left adjoint to the functor taking a monoid to the underlying semigroup.
* '''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-hom adjunction|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<sub>'''Z'''</sub>(''M'',''A'') for every abelian group ''A'', is a right adjoint to ''F''.
* '''From monoids and groups to rings.''' The [[integral monoid ring]] construction gives a functor from [[monoid]]s to rings. This functor is left adjoint to the functor that associates to a given ring its underlying multiplicative monoid. Similarly, the [[integral group ring]] construction yields a functor from [[group (mathematics)|groups]] to rings, left adjoint to the functor that assigns to a given ring its [[group of units]]. One can also start with a [[field (mathematics)|field]] ''K'' and consider the category of ''K''-[[associative algebra|algebras]] instead of the category of rings, to get the monoid and group rings over ''K''.
* '''Field of fractions.''' Consider the category '''Dom'''<sub>m</sub> of integral domains with injective morphisms. The forgetful functor '''Field''' → '''Dom'''<sub>m</sub> from fields has a left adjoint—it assigns to every integral domain its [[field of fractions]].
* '''Polynomial rings'''. Let '''Ring'''<sub>*</sub> be the category of pointed commutative rings with unity (pairs (A,a) where A is a ring, a ∈ A and morphisms preserve the distinguished elements). The forgetful functor G:'''Ring'''<sub>*</sub> → '''Ring''' has a left adjoint – it assigns to every ring R the pair (R[x],x) where R[x] is the [[polynomial ring]] with coefficients from R.
* '''Abelianization'''. Consider the inclusion functor ''G'' : '''Ab''' → '''Grp''' from the [[category of abelian groups]] to [[category of groups]]. It has a left adjoint called [[abelianization]] which assigns to every group ''G'' the quotient group ''G''<sup>ab</sup>=''G''/[''G'',''G''].
* '''The Grothendieck group'''. In [[K-theory]], the point of departure is to observe that the category of [[vector bundle]]s on a [[topological space]] has a commutative monoid structure under [[Direct sum of modules|direct sum]]. One may make an [[abelian group]] out of this monoid, the [[Grothendieck group]], by formally adding an additive inverse for each bundle (or equivalence class). Alternatively one can observe that the functor that for each group takes the underlying monoid (ignoring inverses) has a left adjoint. This is a once-for-all construction, in line with the third section discussion above. That is, one can imitate the construction of [[negative number]]s; but there is the other option of an [[existence theorem]]. For the case of finitary algebraic structures, the existence by itself can be referred to [[universal algebra]], or [[model theory]]; naturally there is also a proof adapted to category theory, too.
* '''Frobenius reciprocity''' in the [[group representation|representation theory of groups]]: see [[induced representation]]. This example foreshadowed the general theory by about half a century.

====Topology====
* '''A functor with a left and a right adjoint.''' Let ''G'' be the functor from [[topological space]]s to [[Set (mathematics)|sets]] that associates to every topological space its underlying set (forgetting the topology, that is). ''G'' has a left adjoint ''F'', creating the [[discrete space]] on a set ''Y'', and a right adjoint ''H'' creating the [[trivial topology]] on ''Y''.
* '''Suspensions and loop spaces.'''  Given [[topological spaces]] ''X'' and ''Y'', the space [''SX'', ''Y''] of [[homotopy classes]] of maps from the [[suspension (topology)|suspension]] ''SX'' of ''X'' to ''Y'' is naturally isomorphic to the space [''X'', Ω''Y''] of homotopy classes of maps from ''X'' to the [[loop space]] Ω''Y'' of ''Y''.  The suspension functor is therefore left adjoint to the loop space functor in the [[homotopy category]], an important fact in [[homotopy theory]].
* '''Stone–Čech compactification.''' Let '''KHaus''' be the category of [[compact space|compact]] [[Hausdorff space]]s and ''G'' : '''KHaus''' → '''Top''' be the inclusion functor to the category of [[topological spaces]]. Then ''G'' has a left adjoint ''F'' : '''Top''' → '''KHaus''', the [[Stone–Čech compactification]]. The unit of this adjoint pair yields a [[continuous function (topology)|continuous]] map from every topological space ''X'' into its Stone–Čech compactification. 
* '''Direct and inverse images of sheaves.''' Every [[continuous map]] ''f'' : ''X'' → ''Y'' between [[topological space]]s induces a functor ''f''<sub> ∗</sub> from the category of [[sheaf (mathematics)|sheaves]] (of sets, or abelian groups, or rings...) on ''X'' to the corresponding category of sheaves on ''Y'', the ''[[direct image functor]]''. It also induces a functor ''f''{{i sup|−1}} from the category of sheaves of abelian groups on ''Y'' to the category of sheaves of abelian groups on ''X'', the ''[[inverse image functor]]''. ''f''{{i sup|−1}} is left adjoint to ''f''<sub> ∗</sub>. Here a more subtle point is that the left adjoint for [[coherent sheaf|coherent sheaves]] will differ from that for sheaves (of sets).
* '''Soberification.''' The article on [[Stone duality]] describes an adjunction between the category of topological spaces and the category of [[sober space]]s that is known as soberification. Notably, the article also contains a detailed description of another adjunction that prepares the way for the famous [[duality (category theory)|duality]] of sober spaces and spatial locales, exploited in [[pointless topology]].

====Posets====
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 operator]]s and to inverse order-preserving bijections between the corresponding closed elements.

As is the case for [[Galois group]]s, the real interest lies often in refining a correspondence to a [[duality (mathematics)|duality]] (i.e. ''antitone'' order isomorphism). A treatment of Galois theory along these lines by [[Irving Kaplansky|Kaplansky]] was influential in the recognition of the general structure here.

The partial order case collapses the adjunction definitions quite noticeably, but can provide several themes:
* adjunctions may not be dualities or isomorphisms, but are candidates for upgrading to that status
* closure operators may indicate the presence of adjunctions, as corresponding [[monad (category theory)|monads]] (cf. the [[Kuratowski closure axioms]])
* a very general comment of [[William Lawvere]]<ref>[[William Lawvere|Lawvere, F. William]], "[http://www.tac.mta.ca/tac/reprints/articles/16/tr16abs.html Adjointness in foundations]", ''Dialectica'', 1969. The notation is different nowadays; an easier introduction by Peter Smith [http://www.logicmatters.net/resources/pdfs/Galois.pdf in these lecture notes], which also attribute the concept to the article cited.</ref> is that ''syntax and semantics'' are adjoint: take ''C'' to be the set of all logical theories (axiomatizations), and ''D'' the power set of the set of all mathematical structures. For a theory ''T'' in ''C'', let ''G''(''T'') be the set of all structures that satisfy the axioms ''T''; for a set of mathematical structures ''S'', let ''F''(''S'') be the minimal axiomatization of ''S''. We can then say that ''S'' is a subset of ''G''(''T'') if and only if ''F''(''S'') logically implies ''T'': the "semantics functor" ''G'' is right adjoint to the "syntax functor" ''F''.
* [[division (mathematics)|division]] is (in general) the attempt to ''invert'' multiplication, but in situations where this is not possible, we often attempt to construct an ''adjoint'' instead: the [[ideal quotient]] is adjoint to the multiplication by [[ring ideal]]s, and the [[material conditional|implication]] in [[propositional calculus|propositional logic]] is adjoint to [[logical conjunction]].

====Category theory====
* '''Equivalences.''' If ''F'' : ''D'' → ''C'' is an [[equivalence of categories]], then we have an inverse equivalence ''G'' : ''C'' → ''D'', and the two functors ''F'' and ''G'' form an adjoint pair. The unit and counit are natural isomorphisms in this case. If η : id → ''GF'' and ε : ''GF'' → id are natural isomorphisms, then there exist unique natural isomorphisms ε' : ''GF'' → id and η' : id → ''GF'' for which (η, ε') and (η', ε) are counit–unit pairs for ''F'' and ''G''; they are
*:<math>\varepsilon'=\varepsilon\circ(F\eta^{-1}G)\circ(FG\varepsilon^{-1})</math>
*:<math>\eta'=(GF\eta^{-1})\circ(G\varepsilon^{-1}F)\circ\eta</math>
* '''A series of adjunctions.''' The functor π<sub>0</sub> which assigns to a category its set of connected components is left-adjoint to the functor ''D'' which assigns to a set the discrete category on that set.  Moreover, ''D'' is left-adjoint to the object functor ''U'' which assigns to each category its set of objects, and finally ''U'' is left-adjoint to ''A'' which assigns to each set the indiscrete category<ref>{{cite web |title=Indiscrete category |url=http://ncatlab.org/nlab/show/indiscrete+category |website=nLab}}</ref> on that set.
* '''Exponential object'''. In a [[cartesian closed category]] the endofunctor ''C'' → ''C'' given by –×''A'' has a right adjoint –<sup>''A''</sup>. This pair is often referred to as [[currying]] and uncurrying; in many special cases, they are also continuous and form a homeomorphism.
<!--* '''Limits and Colimits.''' Limits and colimits can actually be viewed using adjoints when looking at functor categories. If C and D are two categories, then the functor '''limit''' from the category of functors from C to D to the category of constant functors from C to D which takes a given functor from C to D to its limit is in fact right-adjoint to the forgetful functor from the category of constant functors from C to D to the category of functors from C to D. Colimit is similarly the left-adjoint of this forgetful functor from the category of constant functors from C to D to the category of functors from C to D. -->

====Categorical logic====
* '''Quantification.''' If <math>\phi_Y</math> is a unary predicate expressing some property, then a sufficiently strong set theory may prove the existence of the set <math>Y=\{y\mid\phi_Y(y)\}</math> of terms that fulfill the property. A proper subset <math>T\subset Y</math> and the associated injection of <math>T</math> into <math>Y</math> is characterized by a predicate <math>\phi_T(y)=\phi_Y(y)\land\varphi(y)</math> expressing a strictly more restrictive property. 
:The role of [[Quantifier (logic)|quantifiers]] in predicate logics is in forming propositions and also in expressing sophisticated predicates by closing formulas with possibly more variables. For example, consider a predicate <math>\psi_f</math> with two open variables of sort <math>X</math> and <math>Y</math>. Using a quantifier to close <math>X</math>, we can form the set
::<math>\{y\in Y\mid \exists x.\,\psi_f(x,y)\land\phi_{S}(x)\}</math>
:of all elements <math>y</math> of <math>Y</math> for which there is an <math>x</math> to which it is <math>\psi_f</math>-related, and which itself is characterized by the property <math>\phi_{S}</math>. Set theoretic operations like the intersection <math>\cap</math> of two sets directly corresponds to the conjunction <math>\land</math> of predicates. In [[categorical logic]], a subfield of [[topos theory]], quantifiers are identified with adjoints to the pullback functor. Such a realization can be seen in analogy to the discussion of propositional logic using set theory but the general definition make for a richer range of logics.

:So consider an object <math>Y</math> in a category with pullbacks. Any morphism <math>f:X\to Y</math> induces a functor 
::<math>f^{*} : \text{Sub}(Y) \longrightarrow \text{Sub}(X)</math> 
:on the category that is the preorder of [[subobject | subobjects]]. It maps subobjects <math>T</math> of <math>Y</math> (technically: monomorphism classes of <math>T\to Y</math>) to the pullback <math>X\times_Y T</math>. If this functor has a left- or right adjoint, they are called <math>\exists_f</math> and <math>\forall_f</math>, respectively.<ref>[[Saunders Mac Lane|Mac Lane, Saunders]]; Moerdijk, Ieke (1992) ''Sheaves in Geometry and Logic'', Springer-Verlag. {{ISBN|0-387-97710-4}} ''See page 58''</ref> They both map from <math>\text{Sub}(X)</math> back to <math>\text{Sub}(Y)</math>. Very roughly, given a domain <math>S\subset X</math> to quantify a relation expressed via <math>f</math> over, the functor/quantifier closes <math>X</math> in <math>X\times_Y T</math> and returns the thereby specified subset of <math>Y</math>.

: '''Example''': In <math>\operatorname{Set}</math>, the category of sets and functions, the canonical subobjects are the subset (or rather their canonical injections). The pullback <math>f^{*}T=X\times_Y T</math> of an injection of a subset <math>T</math> into <math>Y</math> along <math>f</math> is characterized as the largest set which knows all about <math>f</math> and the injection of <math>T</math> into <math>Y</math>. It therefore turns out to be (in bijection with) the inverse image <math>f^{-1}[T]\subseteq X</math>.
:For <math>S \subseteq X</math>, let us figure out the left adjoint, which is defined via
::<math>{\operatorname{Hom}}(\exists_f S,T) 
\cong 
{\operatorname{Hom}}(S,f^{*}T),</math>
:which here just means
::<math>\exists_f S\subseteq T
\leftrightarrow 
S\subseteq f^{-1}[T]</math>.

:Consider <math> f[S] \subseteq T </math>. We see <math>S\subseteq f^{-1}[f[S]]\subseteq f^{-1}[T]</math>. Conversely, If for an <math>x\in S</math> we also have <math>x\in f^{-1}[T]</math>, then clearly <math> f(x)\in T </math>. So <math> S \subseteq f^{-1}[T] </math> implies <math> f[S] \subseteq T </math>. We conclude that left adjoint to the inverse image functor <math>f^{*}</math> is given by the direct image. Here is a characterization of this result, which matches more the logical interpretation: The image of <math>S</math> under <math>\exists_f </math> is the full set of <math>y</math>'s, such that <math> f^{-1} [\{y\}] \cap S</math> is non-empty. This works because it neglects exactly those <math>y\in Y</math> which are in the complement of <math>f[S]</math>. So
::<math>
\exists_f S 
= \{ y \in Y \mid \exists (x \in f^{-1}[\{y\}]).\, x \in S \; \}
= f[S].
</math>
:Put this in analogy to our motivation <math>\{y\in Y\mid\exists x.\,\psi_f(x,y)\land\phi_{S}(x)\}</math>.
:The right adjoint to the inverse image functor is given (without doing the computation here) by
::<math>
\forall_f S 
= \{ y \in Y \mid \forall (x \in f^{-1} [\{y\}]).\, x \in S \; \}.
</math>
: The subset <math>\forall_f S</math> of <math>Y</math> is characterized as the full set of <math>y</math>'s with the property that the inverse image of <math>\{y\}</math> with respect to <math>f</math> is fully contained within <math>S</math>. Note how the predicate determining the set is the same as above, except that <math>\exists</math> is replaced by <math>\forall</math>.

:''See also [[powerset]].''

==== Probability ====
The twin fact in probability can be understood as an adjunction: that expectation commutes with affine transform, and that the expectation is in some sense the best ''solution'' to the problem of finding a real-valued approximation to a distribution on the real numbers.

Define a category based on <math>\R</math>, with objects being the real numbers, and the morphisms being "affine functions evaluated at a point". That is, for any affine function <math>f(x) = ax + b</math> and any real number <math>r</math>, define a morphism <math>(r, f): r \to f(r)</math>.

Define a category based on <math>M(\R)</math>, the set of probability distribution on <math>\R</math> with finite expectation. Define morphisms on <math>M(\R)</math> as "affine functions evaluated at a distribution". That is, for any affine function <math>f(x) = ax + b</math> and any <math>\mu\in M(\R)</math>, define a morphism <math>(\mu, f): \mu \to \mu\circ f^{-1}</math>.

Then, the [[Dirac delta measure]] defines a functor: <math>\delta: x\mapsto \delta_x</math>, and the expectation defines another functor <math>\mathbb E: \mu \mapsto \mathbb E[\mu]</math>, and they are adjoint: <math>\mathbb E \dashv \delta</math>. (Somewhat disconcertingly, <math>\mathbb E</math> is the left adjoint, even though <math>\mathbb E</math> is "forgetful" and <math>\delta</math> is "free".)