Representable Functors

Some functors only look like they build something new: they are the arrows out of one fixed object, which is what makes forgetting structure the same as collecting elements.

The idea

A representable functor is a set-valued functor whose values are, up to natural isomorphism, the arrows out of one fixed object — or into it.

The forgetful functor $\mathbf{Grp} \to \mathbf{Set}$ is the instance to hold on to. It appears to destroy information: hand it a group, and it hands back the bare set of elements. But a homomorphism from $\mathbb{Z}$ to a group $G$ is decided by where the generator goes, and the generator may go anywhere, so the homomorphisms $\mathbb{Z} \to G$ are the elements of $G$. Forgetting does not throw structure away; it looks at $G$ from $\mathbb{Z}$.

Every object of a locally small category — one whose collections of arrows $\mathcal{C}(A,B)$ are sets — provides such a view. The covariant hom-functor $H^{A} = \mathcal{C}(A,-) \colon \mathcal{C} \to \mathbf{Set}$ sends an object $B$ to the set $\mathcal{C}(A,B)$ and an arrow $g \colon B \to B'$ to post-composition $p \mapsto g \circ p$. Its contravariant partner $H_{A} = \mathcal{C}(-,A)$ sends $B$ to $\mathcal{C}(B,A)$ and an arrow to precomposition, which runs the other way; it records not what $A$ sees but how $A$ is seen.

Definition (Representable functor).

A functor $X \colon \mathcal{C} \to \mathbf{Set}$ is representable when it is naturally isomorphic to $H^{A} = \mathcal{C}(A,-)$ for some object $A$ of $\mathcal{C}$, and a functor $\mathcal{C}^{\mathrm{op}} \to \mathbf{Set}$ is representable when it is naturally isomorphic to some $H_{A} = \mathcal{C}(-,A)$. The object $A$ is then a representing object for the functor.

A functor $X \colon \mathcal{C} \to \mathbf{Set}$ hands over a set of things at each object; calling $X$ representable says the things were arrows all along, all of them leaving one fixed object. The forgetful functor above is represented by $\mathbb{Z}$.

Ways to work on it

• Walkthrough. The hom-functors C(A,-) and C(-,A), what representable means, and the standard representing objects.
• Proof. Why a representable functor determines its representing object up to isomorphism, by feeding the identity arrow to a natural isomorphism.
• Practice. Name the representing object of a forgetful functor, count a hom-set out of a fixed group, and decide whether a functor on Set is representable.
• Hardest. Decide whether harder functors are representable, and count the natural transformations between two hom-functors.

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