Representable functor

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

F \;\cong\; \mathrm{Hom}_{\mathcal{C}}(A,-).

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

F \;\cong\; \mathrm{Hom}_{\mathcal{C}}(-,A),

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 $1$ :

\mathrm{Hom}_{\mathbf{Set}}(1,X)\cong X,

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):

\mathrm{Hom}_{\mathbf{Grp}}(\mathbb{Z},G)\cong U(G),

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 $R$ :

\mathrm{Hom}_{R}(R,M)\cong U(M),

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

Example (Set: the identity functor)

Example (Grp: the forgetful functor)

Example (RRR-Mod: the forgetful functor)

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