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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, particularly
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets and functions) 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 sets in posets, and of Cayley's theorem in
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
.


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'',–) 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 In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
to Hom(''A'',–) for some object ''A'' of C. A representation of ''F'' is a pair (''A'', Φ) where :Φ : Hom(''A'',–) → ''F'' is a natural isomorphism. A contravariant functor ''G'' from C to Set is the same thing as a functor ''G'' : Cop → Set and is commonly called a presheaf. A presheaf is representable when it is naturally isomorphic to the contravariant hom-functor Hom(–,''A'') for some object ''A'' of C.


Universal elements

According to Yoneda's lemma, natural transformations from Hom(''A'',–) to ''F'' are in one-to-one correspondence with the elements of ''F''(''A''). Given a natural transformation Φ : Hom(''A'',–) → ''F'' the corresponding element ''u'' ∈ ''F''(''A'') is given by :u = \Phi_A(\mathrm_A).\, Conversely, given any element ''u'' ∈ ''F''(''A'') we may define a natural transformation Φ : Hom(''A'',–) → ''F'' via :\Phi_X(f) = (Ff)(u)\, 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 → Set is a pair (''A'',''u'') consisting of an object ''A'' of C and an element ''u'' ∈ ''F''(''A'') such that for every pair (''X'',''v'') consisting of an object ''X'' of C and an element ''v'' ∈ ''F''(''X'') there exists a unique morphism ''f'' : ''A'' → ''X'' such that (''Ff'')(''u'') = ''v''. A universal element may be viewed as a
universal morphism In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently f ...
from the one-point set to the functor ''F'' or as an initial object in the
category of elements In category theory, if is a category and is a set-valued functor, the category of elements of (also denoted ) is the following category: * Objects are pairs (A,a) where A \in \mathop(C) and a \in FA. * Morphisms (A,a) \to (B,b) are arrows f: A ...
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

* 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''X''(''f'') = (''Pf'')''u'' = ''f''−1(''u''). Take ''A'' = and ''u'' = . Given a subset ''S'' ⊆ ''X'' the corresponding function from ''X'' to ''A'' is the characteristic function of ''S''. * Forgetful functors 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 mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
in one variable with
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
coefficients. ** The forgetful functor Vect → Set on the
category of real vector spaces In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ring ...
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 ''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. The unique hom-functor Hom(•,–) 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 ''G''-torsor or heap). Choosing a representation amounts to choosing an identity for the heap. *Let ''C'' be the category of CW-complexes with morphisms given by homotopy classes of continuous functions. For each natural number ''n'' there is a contravariant functor ''H''''n'' : ''C'' → Ab which assigns each CW-complex its ''n''th 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 mathematics, Brown's representability theorem in homotopy theory gives necessary and sufficient conditions for a contravariant functor ''F'' on the homotopy category ''Hotc'' of pointed connected CW complexes, to the category of sets Set, to be ...
in algebraic topology says that this functor is represented by a CW-complex ''K''(Z,''n'') called an Eilenberg–MacLane space. *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'' ⊗''R'' ''N''.


Properties


Uniqueness

Representations of functors are unique up to a unique isomorphism. That is, if (''A''11) and (''A''22) represent the same functor, then there exists a unique isomorphism φ : ''A''1 → ''A''2 such that :\Phi_1^\circ\Phi_2 = \mathrm(\varphi,-) as natural isomorphisms from Hom(''A''2,–) to Hom(''A''1,–). This fact follows easily from Yoneda's lemma. Stated in terms of universal elements: if (''A''1,''u''1) and (''A''2,''u''2) represent the same functor, then there exists a unique isomorphism φ : ''A''1 → ''A''2 such that :(F\varphi)u_1 = u_2.


Preservation of limits

Representable functors are naturally isomorphic to Hom functors and therefore share their properties. In particular, (covariant) representable functors 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''X''(•)) 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 copowers 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 In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently f ...
s and adjoint functors 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 In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
(''A'',φ) is a representation of the functor Hom''C''(''X'',''G''–) from ''D'' to Set. It follows that ''G'' has a left-adjoint ''F'' if and only if Hom''C''(''X'',''G''–) is representable for all ''X'' in ''C''. The natural isomorphism Φ''X'' : Hom''D''(''FX'',–) → Hom''C''(''X'',''G''–) yields the adjointness; that is :\Phi_\colon \mathrm_(FX,Y) \to \mathrm_(X,GY) 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''D''(''F''–,''Y'') from ''C'' to Set. It follows that ''F'' has a right-adjoint ''G'' if and only if Hom''D''(''F''–,''Y'') is representable for all ''Y'' in ''D''.


See also

* Subobject classifier * Density theorem


References

* {{Functors