Editing Representable functor

{{Short description|Functor type}}
In [[mathematics]], particularly [[category theory]], a '''representable functor''' is a certain [[functor]] from an arbitrary [[category (mathematics)|category]] into the [[category of sets]]. Such functors give representations of an abstract category in terms of known structures (i.e. [[Set (mathematics)|sets]] and [[function (mathematics)|function]]s) allowing one to utilize, as much as possible, knowledge about the category of sets in other settings.

From another point of view, representable functors for a category ''C'' are the functors ''given'' with ''C''. Their theory is a vast generalisation of [[upper set]]s in [[poset]]s, and [[Yoneda lemma|Yoneda's representability]] theorem generalizes [[Cayley's theorem]] in [[group theory]].

==Definition==

Let '''C''' be a [[locally small category]] and let '''Set''' be the [[category of sets]]. For each object ''A'' of '''C''' let Hom(''A'',&ndash;) be the [[hom functor]] that maps object ''X'' to the set Hom(''A'',''X'').

A [[functor]] ''F'' : '''C''' → '''Set''' is said to be '''representable''' if it is [[naturally isomorphic]] to Hom(''A'',&ndash;) for some object ''A'' of '''C'''. A '''representation''' of ''F'' is a pair (''A'', Φ) where
:&Phi; : Hom(''A'',&ndash;) &rarr; ''F''
is a natural isomorphism.

A [[contravariant functor]] ''G'' from '''C''' to '''Set''' is the same thing as a functor ''G'' : '''C'''<sup>op</sup> → '''Set''' and is commonly called a [[presheaf (category theory)|presheaf]]. A presheaf is representable when it is naturally isomorphic to the contravariant hom-functor Hom(&ndash;,''A'') for some object ''A'' of '''C'''.

== Universal elements ==
According to [[Yoneda's lemma]], natural transformations from Hom(''A'',&ndash;) to ''F'' are in one-to-one correspondence with the elements of ''F''(''A''). Given a natural transformation Φ : Hom(''A'',&ndash;) → ''F'' the corresponding element ''u'' ∈ ''F''(''A'') is given by
:<math>u = \Phi_A(\mathrm{id}_A).\,</math>
Conversely, given any element ''u'' ∈ ''F''(''A'') we may define a natural transformation Φ : Hom(''A'',&ndash;) → ''F'' via
:<math>\Phi_X(f) = (Ff)(u)\,</math>
where ''f'' is an element of Hom(''A'',''X''). In order to get a representation of ''F'' we want to know when the natural transformation induced by ''u'' is an isomorphism. This leads to the following definition:
:A '''universal element''' of a functor ''F'' : '''C''' &rarr; '''Set''' is a pair (''A'',''u'') consisting of an object ''A'' of '''C''' and an element ''u'' &isin; ''F''(''A'') such that for every pair (''X'',''v'') consisting of an object ''X'' of '''C''' and an element ''v'' &isin; ''F''(''X'') there exists a unique morphism ''f'' : ''A'' &rarr; ''X'' such that (''Ff'')(''u'') = ''v''.
A universal element may be viewed as a [[universal morphism]] from the one-point set {•} to the functor ''F'' or as an [[initial object]] in the [[category of elements]] of ''F''.

The natural transformation induced by an element ''u'' ∈ ''F''(''A'') is an isomorphism if and only if (''A'',''u'') is a universal element of ''F''. We therefore conclude that representations of ''F'' are in one-to-one correspondence with universal elements of ''F''. For this reason, it is common to refer to universal elements (''A'',''u'') as representations.

==Examples==

* The [[functor represented by a scheme]] ''A'' can sometimes describe families of geometric objects''.'' For example, [[Vector bundle|vector bundles]] of rank ''k'' over a given algebraic variety or scheme ''X'' correspond to algebraic morphisms <math>X\to A</math> where ''A'' is the [[Grassmannian]] of ''k''-planes in a high-dimensional space. Also certain types of subschemes are represented by [[Hilbert scheme|Hilbert schemes]]. 
* Let ''C'' be the category of [[CW-complex]]es with morphisms given by homotopy classes of continuous functions. For each natural number ''n'' there is a contravariant functor ''H''<sup>''n''</sup> : ''C'' → '''Ab''' which assigns each CW-complex its ''n''<sup>th</sup> [[cohomology group]] (with integer coefficients). Composing this with the [[forgetful functor]] we have a contravariant functor from ''C'' to '''Set'''. [[Brown's representability theorem]] in algebraic topology says that this functor is represented by a CW-complex ''K''('''Z''',''n'') called an [[Eilenberg–MacLane space]].
*Consider the contravariant functor ''P'' : '''Set''' → '''Set''' which maps each set to its [[power set]] and each function to its [[inverse image]] map. To represent this functor we need a pair (''A'',''u'') where ''A'' is a set and ''u'' is a subset of ''A'', i.e. an element of ''P''(''A''), such that for all sets ''X'', the hom-set Hom(''X'',''A'') is isomorphic to ''P''(''X'') via Φ<sub>''X''</sub>(''f'') = (''Pf'')''u'' = ''f''<sup>−1</sup>(''u''). Take ''A'' = {0,1} and ''u'' = {1}. Given a subset ''S'' ⊆ ''X'' the corresponding function from ''X'' to ''A'' is the [[indicator function|characteristic function]] of ''S''.
*[[Forgetful functor]]s to '''Set''' are very often representable. In particular, a forgetful functor is represented by (''A'', ''u'') whenever ''A'' is a [[free object]] over a [[singleton set]] with generator ''u''.
** The forgetful functor '''Grp''' → '''Set''' on the [[category of groups]] is represented by ('''Z''', 1).
** The forgetful functor '''Ring''' → '''Set''' on the [[category of rings]] is represented by ('''Z'''[''x''], ''x''), the [[polynomial ring]] in one [[Variable (mathematics)|variable]] with [[integer]] [[coefficient]]s.
** The forgetful functor '''Vect''' → '''Set''' on the [[category of real vector spaces]] is represented by ('''R''', 1).
** The forgetful functor '''Top''' → '''Set''' on the [[category of topological spaces]] is represented by any singleton topological space with its unique element.
*A [[group (mathematics)|group]] ''G'' can be considered a category (even a [[groupoid]]) with one object which we denote by •. A functor from ''G'' to '''Set''' then corresponds to a [[G-set|''G''-set]]. The unique hom-functor Hom(•,&ndash;) from ''G'' to '''Set''' corresponds to the canonical ''G''-set ''G'' with the action of left multiplication. Standard arguments from group theory show that a functor from ''G'' to '''Set''' is representable if and only if the corresponding ''G''-set is simply transitive (i.e. a [[torsor|''G''-torsor]] or [[heap (mathematics)|heap]]). Choosing a representation amounts to choosing an identity for the heap.
*Let ''R'' be a commutative ring with identity, and let '''R'''-'''Mod''' be the category of ''R''-modules. If ''M'' and ''N'' are unitary modules over ''R'', there is a covariant functor ''B'': '''R'''-'''Mod''' → '''Set''' which assigns to each ''R''-module ''P'' the set of ''R''-bilinear maps ''M'' × ''N'' → ''P'' and to each ''R''-module homomorphism ''f'' : ''P'' → ''Q'' the function ''B''(''f'') : ''B''(''P'') → ''B''(''Q'') which sends each bilinear map ''g'' : ''M'' × ''N'' → ''P'' to the bilinear map ''f''∘''g'' : ''M'' × ''N''→''Q''. The functor ''B'' is represented by the ''R''-module ''M'' ⊗<sub>''R''</sub> ''N''.<ref>{{cite book|last1=Hungerford|first1=Thomas|title=Algebra|publisher=Springer-Verlag|isbn=3-540-90518-9|page=470}}</ref>

== Analogy: Representable functionals ==
Consider a linear functional on a complex [[Hilbert space]] ''H'', i.e. a linear function <math>F: H\to\mathbb C</math>. The [[Riesz representation theorem]] states that if ''F'' is continuous, then there exists a unique element <math>a\in H</math> which represents ''F'' in the sense that ''F'' is equal to the inner product functional <math>\langle a, -\rangle </math>, that is <math>F(v) = \langle a,v\rangle </math> for <math>v\in H</math>. 

For example, the continuous linear functionals on the [[Square-integrable function|square-integrable function space]] <math>H = L^2(\mathbb R)</math> are all representable in the form <math>\textstyle F(v) = \langle a,v\rangle =  \int_{\mathbb R} a(x)v(x)\,dx</math> for a unique function <math>a(x)\in H</math>. The theory of [[Distribution (mathematics)|distributions]] considers more general continuous functionals on the space of test functions <math>C=C^\infty_c(\mathbb R)</math>. Such a distribution functional is not necessarily representable by a function, but it may be considered intuitively as a generalized function. For instance, the [[Dirac delta function]] is the distribution defined by <math>F(v) = v(0)</math> for each test function <math>v(x)\in C</math>, and may be thought of as "represented" by an infinitely tall and thin bump function near <math>x=0</math>.  

Thus, a function <math>a(x)</math> may be determined not by its values, but by its effect on other functions via the inner product. Analogously, an object ''A'' in a category may be characterized not by its internal features, but by its [[Functor represented by a scheme|functor of points]], i.e. its relation to other objects via morphisms. Just as non-representable functionals are described by distributions, non-representable functors may be described by more complicated structures such as [[Stack (mathematics)|stacks]]. 

==Properties==

===Uniqueness===

Representations of functors are unique up to a unique isomorphism. That is, if (''A''<sub>1</sub>,Φ<sub>1</sub>) and (''A''<sub>2</sub>,Φ<sub>2</sub>) represent the same functor, then there exists a unique isomorphism φ : ''A''<sub>1</sub> → ''A''<sub>2</sub> such that
:<math>\Phi_1^{-1}\circ\Phi_2 = \mathrm{Hom}(\varphi,-)</math>
as natural isomorphisms from Hom(''A''<sub>2</sub>,&ndash;) to Hom(''A''<sub>1</sub>,&ndash;). This fact follows easily from [[Yoneda's lemma]].

Stated in terms of universal elements: if (''A''<sub>1</sub>,''u''<sub>1</sub>) and (''A''<sub>2</sub>,''u''<sub>2</sub>) represent the same functor, then there exists a unique isomorphism φ : ''A''<sub>1</sub> → ''A''<sub>2</sub> such that 
:<math>(F\varphi)u_1 = u_2.</math>

===Preservation of limits===

Representable functors are naturally isomorphic to Hom functors and therefore share their properties. In particular, (covariant) representable functors [[Limit (category theory)#Preservation of limits|preserve all limits]]. It follows that any functor which fails to preserve some limit is not representable.

Contravariant representable functors take colimits to limits.

===Left adjoint===

Any functor ''K'' : ''C'' → '''Set''' with a [[left adjoint]] ''F'' : '''Set''' → ''C'' is represented by (''FX'', η<sub>''X''</sub>(•)) where ''X'' = {•} is a [[singleton set]] and η is the unit of the adjunction.

Conversely, if ''K'' is represented by a pair (''A'', ''u'') and all small [[copower]]s of ''A'' exist in ''C'' then ''K'' has a left adjoint ''F'' which sends each set ''I'' to the ''I''th copower of ''A''.

Therefore, if ''C'' is a category with all small copowers, a functor ''K'' : ''C'' → '''Set''' is representable if and only if it has a left adjoint.

==Relation to universal morphisms and adjoints==

The categorical notions of [[universal morphism]]s and [[adjoint functor]]s can both be expressed using representable functors.

Let ''G'' : ''D'' → ''C'' be a functor and let ''X'' be an object of ''C''. Then (''A'',φ) is a universal morphism from ''X'' to ''G'' [[if and only if]] (''A'',φ) is a representation of the functor Hom<sub>''C''</sub>(''X'',''G''&ndash;) from ''D'' to '''Set'''. It follows that ''G'' has a left-adjoint ''F'' if and only if Hom<sub>''C''</sub>(''X'',''G''&ndash;) is representable for all ''X'' in ''C''. The natural isomorphism Φ<sub>''X''</sub> : Hom<sub>''D''</sub>(''FX'',&ndash;) → Hom<sub>''C''</sub>(''X'',''G''&ndash;) yields the adjointness; that is 
:<math>\Phi_{X,Y}\colon \mathrm{Hom}_{\mathcal D}(FX,Y) \to \mathrm{Hom}_{\mathcal C}(X,GY)</math>
is a bijection for all ''X'' and ''Y''.

The dual statements are also true. Let ''F'' : ''C'' → ''D'' be a functor and let ''Y'' be an object of ''D''. Then (''A'',φ) is a universal morphism from ''F'' to ''Y'' if and only if (''A'',φ) is a representation of the functor Hom<sub>''D''</sub>(''F''&ndash;,''Y'') from ''C'' to '''Set'''. It follows that ''F'' has a right-adjoint ''G'' if and only if Hom<sub>''D''</sub>(''F''&ndash;,''Y'') is representable for all ''Y'' in ''D''.<ref>{{cite book |last=Nourani |first=Cyrus |title=A Functorial Model Theory: Newer Applications to Algebraic Topology, Descriptive Sets, and Computing Categories Topos |date=19 April 2016 |publisher=CRC Press |page=28 |isbn=978-1482231502}}</ref>

== See also ==
* [[Subobject classifier]]
* [[Density theorem (category theory)|Density theorem]]

== References ==
<references />

*{{cite book | first = Saunders | last = Mac Lane | author-link = Saunders Mac Lane | year = 1998 | title = [[Categories for the Working Mathematician]] | series = Graduate Texts in Mathematics '''5''' | edition = 2nd | publisher = Springer | isbn = 0-387-98403-8}}

{{Functors}}

[[Category:Representable functors| ]]