Hom-Functors and Representability

Core Idea

For an object A in a category C, the contravariant hom-functor Hom(−, A): C^op → Set is a fundamental example of a set-valued functor. A functor F: C → Set is representable if it is naturally isomorphic to Hom(−, A) for some object A. Representability is equivalent to the existence of a universal element, and the Yoneda lemma characterizes all natural transformations from representable functors as evaluations at elements of the representing object.

How It's Best Learned

Study representable functors in Set (where Hom(1, −) ≅ identity), Group (where Hom(Z, −) ≅ identity), and Vec_k. Use the Yoneda lemma to show that any natural transformation between representable functors corresponds uniquely to an element of the representing object.

Common Misconceptions

Not every set-valued functor is representable—representability is a strong condition requiring a universal element. A functor can be 'almost' representable but fail on a single object or natural transformation. Representability depends on the target category (Set vs other categories give different notions).

Explainer

From the Yoneda lemma and representable functors, you already know that for each object A in a category C, the assignment X ↦ Hom(A, X) defines a functor C → Set — the covariant hom-functor Hom(A, −). Similarly, X ↦ Hom(X, A) defines the contravariant hom-functor Hom(−, A): C^op → Set. These hom-functors are the canonical examples of set-valued functors, and every other set-valued functor is judged by comparison to them.

A functor F: C → Set is representable if it is naturally isomorphic to Hom(A, −) for some object A. Spelled out: there exists an object A and a natural isomorphism α: Hom(A, −) ⇒ F, meaning for every object X, there is a bijection αₓ: Hom(A, X) → F(X), and these bijections are compatible with morphisms. The object A is the representing object and is unique up to unique isomorphism (since representability is a universal property). The Yoneda lemma then makes this precise: the natural transformations from Hom(A, −) to any functor F are in bijection with elements of F(A) — natural isomorphisms correspond to distinguished elements that generate all of F(A) naturally.

The key concept is the universal element: u ∈ F(A) is universal if for every object X and every element x ∈ F(X), there exists a unique morphism f: A → X such that F(f)(u) = x. Representability is equivalent to the existence of a universal element. Think of u as the "free" or "generic" element — every other element of F anywhere in the category is uniquely determined by "where u gets sent" under some morphism out of A. This is the categorical version of "generated by one element with no relations." Concrete examples: in Grp, the functor U: Grp → Set (underlying set) is represented by ℤ, the free group on one generator, with universal element the generator 1 ∈ ℤ — every group element corresponds to a unique homomorphism out of ℤ. In CRing, the polynomial ring k[x] represents the "evaluate-at-a-point" functor, with universal element x itself.

Representability is a powerful organizational principle because universal properties are stable under categorical constructions: limits, colimits, adjunctions, and Kan extensions all interact cleanly with representable functors. The Yoneda embedding — the functor C → [C^op, Set] sending A to Hom(−, A) — is fully faithful, meaning C embeds into its presheaf category with no information loss. This gives a precise sense in which any category can be studied by how its objects relate to all other objects via morphisms. Recognizing that a functor is representable means you have found a universal property, and universal properties are the language in which category theory transfers theorems across different mathematical settings without rewriting proofs.