# Representable functor

A covariant (or contravariant) functor $F$ from some category $\mathfrak R$ into the category of sets $\mathfrak S$( cf. Sets, category of) that is isomorphic to one of the functors

$$H ( A, -) : \mathfrak R \rightarrow \mathfrak S ,\ \ X \mapsto H( A, X),$$

or

$$H(-, A): \mathfrak R \rightarrow \mathfrak S ,\ \ X \mapsto H( X, A) .$$

A functor $F: \mathfrak R \rightarrow \mathfrak S$ is representable if and only if there is an object $A \in \mathop{\rm Ob} \mathfrak R$ and an element $a \in F ( A)$ such that for every element $x \in F ( X)$, $X \in \mathop{\rm Ob} \mathfrak R$, there is a unique morphism $\alpha : A \rightarrow X$ for which $x = F ( \alpha )( a)$. The object $A$ is called a representing object for $F$; it is unique up to isomorphism.

In the category of sets the identity functor is representable: a representing object is a singleton. The functor of taking a Cartesian power is also representable: a representing object is a set whose cardinality equals the given power. In an arbitrary category a product of representable functors $F _ {i}$ with representing objects $A _ {i}$, $i \in I$, is representable if and only if the coproduct of the $A _ {i}$ exists in the category. Every covariant representable functor commutes with limits, i.e. is continuous (cf. Continuous functor).

A representable functor is an analogue of the concept of a "free universal algebra with one generator" . For any functor $G: \mathfrak R \rightarrow \mathfrak S$ and a representable functor $F$ the set of natural transformations $\mathop{\rm Nat} ( F, G)$ is isomorphic to $G ( A)$, where $A$ is a representing object for $F$. This shows that representable functors are free objects in the category of functors.

In the case of additive categories one considers additive functors with values in the category of Abelian groups instead of functors with values in $\mathfrak S$. Therefore, in this case one understands a representable functor to be an additive functor isomorphic to one of the form $H( A, -)$ or $H(-, A)$.

The concept of a representable functor arose first in algebraic geometry (cf. ). The most important examples of representable functors in this branch are Picard functors $\mathop{\rm Pic} X/S$ and Hilbert functors $\mathop{\rm Hilb} X/S$, which are representable in the category of algebraic spaces (cf.  and Algebraic space). Let $K$ be the field of fractions of a regular discretely-normed ring $O$ with perfect field of residues. If $X _ {O}$ is a smooth geometrically non-degenerate singular curve of genus $g > 0$ over $K$, then its minimal model represents the functor $Y \mapsto \mathop{\rm Isom} _ {K} ( Y \otimes _ {O} K, X _ {O} )$ from the category of regular $O$- schemes. If $A$ is an Abelian variety over $K$, then its minimal Néron model is a smooth group scheme $X \rightarrow \mathop{\rm Spec} O$, representing the functor $Y \mapsto \mathop{\rm Hom} _ {K} ( Y \otimes _ {O} K, A)$ from the category of smooth $O$- schemes.

How to Cite This Entry:
Representable functor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Representable_functor&oldid=48516
This article was adapted from an original article by S.G. TankeevM.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article