Editing Adjoint functors (section)

==Examples ==
===Free groups===
The construction of [[free group]]s is a common and illuminating example.

Let ''F'' : '''[[category of sets|Set]]''' → '''[[category of groups|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''.  Let <math>\eta_Y:Y\to GFY</math> 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 <math>\eta_Y:Y\to GFY</math> via a unique group homomorphism from ''FY'' to ''W''.  This is precisely the [[Free group#Universal property|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''. Let <math>\varepsilon_X:FGX\to X</math> be the group homomorphism that sends the generators of ''FGX'' to the elements of ''X'' they correspond to, which exists by the universal property of free groups. Then each <math>(GX,\varepsilon_X)</math> is a terminal morphism from ''F'' to ''X'', because any group homomorphism from a free group ''FZ'' to ''X'' will factor through <math>\varepsilon_X:FGX\to X</math> via a unique set map from ''Z'' to ''GX''.  This means that (''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.  Then, a direct verification that they form a counit–unit adjunction <math>(\varepsilon,\eta):F\dashv G</math> is as follows:

; The first counit–unit equation : <math>1_F = \varepsilon F\circ F\eta</math> says that for each set ''Y'' the composition
::<math>FY\xrightarrow\overset{}{\;F(\eta_Y)\;}FGFY\xrightarrow{\;\varepsilon_{FY}\,}FY</math>
: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 <math>F(\eta_Y)</math> 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 <math>\varepsilon_{FY}</math> 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 : <math>1_G = G\varepsilon \circ \eta G</math> says that for each group ''X'' the composition
::<math>GX\xrightarrow{\;\eta_{GX}\;}GFGX\xrightarrow\overset{}{\;G(\varepsilon_X)\,}GX</math>&nbsp;
:should be the identity.  The intermediate set ''GFGX'' is just the underlying set of ''FGX''.  The arrow <math>\eta_{GX}</math> is the "inclusion of generators" set map from the set ''GX'' to the set ''GFGX''.  The arrow <math>G(\varepsilon_X)</math> 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 object]]s are all examples of a left adjoint to a [[forgetful functor]], which assigns to an algebraic object its underlying set.  These algebraic [[free functor]]s have generally the same description as in the detailed description of the free group situation above.

===Diagonal functors and limits===
[[Product (category theory)|Products]], [[Pullback (category theory)|fibred products]], [[Equalizer (mathematics)|equalizers]], and [[Kernel (algebra)|kernels]] are all examples of the categorical notion of a [[limit (category theory)|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<sup>2</sup>''' → '''Grp''' be the functor that assigns to each pair (''X''<sub>1</sub>, ''X<sub>2</sub>'') the product group ''X''<sub>1</sub>×''X''<sub>2</sub>, and let Δ : '''Grp →''' '''Grp<sup>2</sup>'''  be the [[diagonal functor]] that assigns to every group ''X'' the pair (''X'', ''X'') in the product category '''Grp<sup>2</sup>'''. 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 ''X''<sub>1</sub>×''X''<sub>2</sub> to ''X''<sub>1</sub> and ''X''<sub>2</sub> 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 [[Set (mathematics)|sets]], the product of rings, the [[product topology|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 ''f''<sub>1</sub> : ''A''<sub>1</sub> → ''B''<sub>1</sub> and ''f''<sub>2</sub> : ''A''<sub>2</sub> → ''B''<sub>2</sub> are two objects of ''D'', then a morphism from ''f''<sub>1</sub> to ''f''<sub>2</sub> is a pair (''g''<sub>''A''</sub>, ''g''<sub>''B''</sub>) of morphisms such that ''g''<sub>''B''</sub>''f''<sub>1</sub> = ''f''<sub>2</sub>''g''<sub>''A''</sub>. Let ''G'' : ''D'' → '''Ab''' be the functor which assigns to each homomorphism its [[kernel (algebra)|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===
[[Coproduct]]s, [[Pushout (category theory)|fibred coproducts]], [[coequalizer]]s, and [[cokernel]]s are all examples of the categorical notion of a [[limit (category theory)|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.

* '''Coproducts.''' If ''F'' :  '''Ab'''<sup>'''2'''</sup> '''→''' '''Ab''' assigns to every pair (''X''<sub>1</sub>, ''X''<sub>2</sub>) of abelian groups their [[Direct sum of groups|direct sum]], and if ''G'' : '''Ab''' → '''Ab'''<sup>'''2'''</sup> 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 this adjoint pair is the defining pair of inclusion maps from ''X''<sub>1</sub> and ''X''<sub>2</sub> into the direct sum, and the counit is the additive map from the direct sum of (''X'',''X'') to back to ''X'' (sending an element (''a'',''b'') of the direct sum to the element ''a''+''b'' of ''X'').

: Analogous examples are given by the [[Direct sum of modules|direct sum]] of [[vector space]]s and [[module (mathematics)|modules]], by the [[free product]] of groups and by the disjoint union of sets.

===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".)