Representable functor

Definition

Let $\mathcal{C}$ be a category such that each hom-class $\mathrm{Hom}_{\mathcal{C}}(A,B)$ is a set (i.e. $\mathcal{C}$ is locally small).

A covariant functor $F:\mathcal{C}\to \mathbf{Set}$ is representable if there exists an object $A\in \mathcal{C}$ and a natural isomorphism

A contravariant functor $F:\mathcal{C}^{op}\to \mathbf{Set}$ is representable if there exists an object $A\in \mathcal{C}$ and a natural isomorphism

where $\mathcal{C}^{op}$ is the opposite category .

The object $A$ is called a representing object for $F$. If $F$ is representable, then any two representing objects are uniquely isomorphic .

Yoneda viewpoint

By the Yoneda lemma , natural transformations $\mathrm{Hom}_{\mathcal{C}}(A,-)\Rightarrow F$ correspond bijectively to elements of $F(A)$. In particular, a representation $F\cong \mathrm{Hom}_{\mathcal{C}}(A,-)$ is determined by a “universal element” in $F(A)$.

Examples

Example (Set: the identity functor)

In $\mathbf{Set}$, the identity functor $\mathrm{Id}:\mathbf{Set}\to \mathbf{Set}$ is representable by the one-point set $1$:

since a function $1\to X$ is determined by the image of the unique element of $1$. (Here “function” is function .)

Example (Grp: the forgetful functor)

Let $U:\mathbf{Grp}\to \mathbf{Set}$ be the forgetful functor sending a group to its underlying set. Then $U$ is representable by $\mathbb{Z}$ (the free group on one generator):

by sending a homomorphism $\varphi:\mathbb{Z}\to G$ to $\varphi(1)$.

Example ($R$-Mod: the forgetful functor)

Let $U:R\text{-}\mathbf{Mod}\to \mathbf{Set}$ be the forgetful functor. Then $U$ is representable by the regular module $R$:

via $\varphi\mapsto \varphi(1)$. This is natural in $M$.