Representable Functor
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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, cate ...
, a representable functor is a certain
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
from an arbitrary
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
into the
category of sets In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition of m ...
. Such functors give representations of an abstract category in terms of known structures (i.e. sets and
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
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 In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in ''X'') of a partially ordered set (X, \leq) is a subset S \subseteq X with the following property: if ''s'' is in ''S'' and if ''x'' in ''X'' is larger ...
s in
poset In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
s, and of
Cayley's theorem In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group is isomorphic to a subgroup of a symmetric group. More specifically, is isomorphic to a subgroup of the symmetric group \operatorname(G) whose eleme ...
in
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
.


Definition

Let C be a
locally small category In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows asso ...
and let Set be the
category of sets In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition of m ...
. For each object ''A'' of C let Hom(''A'',–) be the
hom functor In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and ...
that maps object ''X'' to the set Hom(''A'',''X''). A
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
''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 In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...
''G'' from C to Set is the same thing as a functor ''G'' : Cop → Set and is commonly called a
presheaf In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
. 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 In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (viewi ...
, 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 fro ...
from the one-point set to the functor ''F'' or as an
initial object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...
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

* Consider the contravariant functor ''P'' : Set → Set which maps each set to its
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
and each function to its
inverse image In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
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 In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function ::\mathbf_A\colon X \to \, :which for a given subset ''A'' of ''X'', has value 1 at points ...
of ''S''. *
Forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given sign ...
s to Set are very often representable. In particular, a forgetful functor is represented by (''A'', ''u'') whenever ''A'' is a
free object In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a set ''A'' can be thought of as being a "generic" algebraic structure over ''A'': the only equations that hold between ele ...
over a
singleton set In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, the ...
with generator ''u''. ** The forgetful functor Grp → Set on the
category of groups In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory. Relation to other categories There a ...
is represented by (Z, 1). ** The forgetful functor Ring → Set on the
category of rings In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, 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 Variable may refer to: * Variable (computer science), a symbolic name associated with a value and whose associated value may be changed * Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...
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 language ...
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
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 In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again contin ...
is represented by any singleton topological space with its unique element. *A
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
''G'' can be considered a category (even a
groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: *''Group'' with a partial functi ...
) 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-complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This clas ...
es 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 In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
(with integer coefficients). Composing this with the
forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given sign ...
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 b ...
in algebraic topology says that this functor is represented by a CW-complex ''K''(Z,''n'') called an
Eilenberg–MacLane space In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. ...
. *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 In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (viewi ...
. 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 In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
''F'' : Set → ''C'' is represented by (''FX'', η''X''(•)) where ''X'' = is a
singleton set In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, the ...
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 fro ...
s and
adjoint functor In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
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 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 bicondi ...
(''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 In category theory, a subobject classifier is a special object Ω of a category such that, intuitively, the subobjects of any object ''X'' in the category correspond to the morphisms from ''X'' to Ω. In typical examples, that morphism assigns "true ...
* Density theorem


References

* {{Functors