In the category Grp, the underlying-set functor U: Grp → Set (which sends a group to its underlying set) is representable, with representing object ℤ. What does this mean concretely?
AEvery group can be generated by a single element, just as ℤ is generated by 1
BThere is a natural bijection between group homomorphisms ℤ → G and elements of the underlying set of G, for every group G
CThe integers ℤ are the initial object in Grp, so every group has a unique map from ℤ
DEvery group is isomorphic to a quotient of ℤ, showing ℤ generates all groups
Representability of U by ℤ means there is a natural isomorphism Hom(ℤ, −) ≅ U. Concretely, for any group G, a group homomorphism f: ℤ → G is completely determined by where the generator 1 ∈ ℤ maps to — and it can map to any element of G. So group homomorphisms from ℤ to G are in natural bijection with elements of G. The universal element is 1 ∈ ℤ (or equivalently, the identity homomorphism id: ℤ → ℤ). This is different from saying every group is cyclic — it's saying that 'morphisms out of ℤ' and 'elements of groups' are the same data.
Question 2 Multiple Choice
A functor F: C → Set has a universal element u ∈ F(A). What does universality of u mean, and why does it imply F is representable?
Au is the largest element in F(A) under some ordering, ensuring F is bounded
BFor every object X and every element x ∈ F(X), there is a unique morphism f: A → X such that F(f)(u) = x — every element of F everywhere is uniquely 'generated' from u
Cu is preserved by all natural transformations from F, making it a fixed point
Du corresponds to the identity morphism on A, which generates all endomorphisms of A
A universal element u ∈ F(A) is one from which every other element of F, in any object X, can be uniquely reached by applying F to a morphism out of A. This means: given any x ∈ F(X), there is a unique f: A → X with F(f)(u) = x. This is exactly what it means to have a natural isomorphism αₓ: Hom(A, X) → F(X) defined by αₓ(f) = F(f)(u) — the morphisms out of A biject with the elements of F(X) naturally. So u ∈ F(A) is universal if and only if F is represented by A with universal element u.
Question 3 True / False
If a functor F: C → Set is representable by object A, then the representing object A is unique up to unique isomorphism.
TTrue
FFalse
Answer: True
Uniqueness up to unique isomorphism is a general property of universal properties. If F is represented by both A and A', then there are natural isomorphisms Hom(A, −) ≅ F ≅ Hom(A', −), which by the Yoneda lemma implies a unique natural isomorphism between Hom(A, −) and Hom(A', −). The Yoneda lemma further implies this natural isomorphism is induced by a unique isomorphism A ≅ A'. This uniqueness is what makes representing objects well-defined mathematical entities — representability pins down the representing object up to unique isomorphism.
Question 4 True / False
Nearly every functor F: C → Set is representable, since for any set-valued functor we can usually construct a suitable hom-functor that matches its values.
TTrue
FFalse
Answer: False
Representability is a strong and rare condition, not a generic property. A functor F: C → Set is representable only if there exists a single object A and a natural isomorphism Hom(A, −) ≅ F. This requires F to have a universal element — a distinguished u ∈ F(A) from which every element of F(X) for every X can be uniquely reached. Many natural functors fail this condition: for example, the functor sending every set to its power set is not representable (no single set has a natural bijection with power sets of all sets). The Yoneda lemma tells us the space of representable functors is exactly the image of the Yoneda embedding, which is only a small slice of all presheaves.
Question 5 Short Answer
Why does the Yoneda lemma imply that recognizing a functor as representable is equivalent to finding a universal property, and why is this organizationally powerful in mathematics?
Think about your answer, then reveal below.
Model answer: The Yoneda lemma states that natural transformations from Hom(A, −) to any functor F are in bijection with elements of F(A). A natural isomorphism Hom(A, −) ≅ F corresponds to a distinguished element u ∈ F(A) that is universal: every element of F(X) for every X is uniquely the image of some morphism A → X applied to u. This means the object A and its universal element u together encode a universal property — a characterization of A by how it maps into everything else, or equivalently, by what it 'freely generates' in F. Finding a universal property for a construction (e.g., 'the free group on one generator', 'the polynomial ring k[x] as universal ring with an element') is equivalent to identifying the representing object. This is organizationally powerful because universal properties are stable under categorical constructions: limits, adjoints, and Kan extensions all preserve representability, so theorems proved once for hom-functors transfer to all representable functors automatically.
This is the practical payoff of category theory: instead of reproing theorems for each mathematical structure separately, recognizing a universal property lets you import results from the general theory. The statement 'ℤ represents the forgetful functor Grp → Set' packages the entire theory of free groups on one generator into a single categorical statement.