Questions: Adjunctions as Natural Hom-set Bijections

Question 1

You want to define a group homomorphism from the free group F({a, b, c}) to the symmetric group S₃. Using the adjunction between Set and Grp, what information is it sufficient to specify?

• A. The images of all elements of F({a, b, c}), listed explicitly, since the homomorphism must be total
• B. The images of the three generators a, b, c in S₃, since the hom-set bijection reduces a group homomorphism to a function on generators
• C. A surjective map from F({a, b, c}) to S₃, ensuring the homomorphism is well-defined
• D. The kernel of the homomorphism, from which all other values can be recovered

Answer: (B)

The adjunction Hom_Grp(F(S), G) ≅ Hom_Set(S, U(G)) says exactly this: a group homomorphism from the free group on S to G is completely determined by a function from the generating set S to the underlying set of G. To define a homomorphism F({a,b,c}) → S₃, you only need to say where a, b, and c go in S₃ — three elements of S₃. The homomorphism then extends uniquely because every element of F({a,b,c}) is a word in a, b, c and their inverses, and homomorphisms must respect the group operations. Option A describes an exponentially larger specification that the universal property makes unnecessary. Options C and D describe different properties and do not follow from the adjunction.

Question 2

In the free-forgetful adjunction between Set and Grp, what does the unit map η_S: S → U(F(S)) represent?

• A. The group multiplication map, defining how elements of F(S) combine
• B. The canonical inclusion of the generating set S into the underlying set of the free group F(S)
• C. The quotient map collapsing F(S) to the trivial group by setting all generators equal
• D. A natural transformation from F to the identity functor on Grp

Answer: (B)

The unit η_S: S → U(F(S)) is the 'canonical inclusion': it sends each element s ∈ S to the corresponding generator word in F(S), viewed as an element of the underlying set. It is the most natural way to embed the original set into the free structure built on it. This is the 'insert generators' map: every element of S appears in F(S) as a generator. The unit's naturality ensures this inclusion behaves coherently with all set functions. Option A confuses the unit with the group operation. Option C describes a collapse, which is the opposite of an inclusion. Option D misidentifies the domain and codomain of η.

Question 3

The naturality condition on the hom-set bijection φ: Hom_D(Lc, d) ≅ Hom_C(c, Rd) ensures that transposing a morphism commutes with pre- and post-composition by arbitrary morphisms.

• T. True
• F. False

Answer: True

Naturality in c means: for any morphism h: c' → c in C, we have φ(f ∘ Lh) = φ(f) ∘ h. Naturality in d means: for any morphism k: d → d' in D, we have φ(k ∘ f) = Rk ∘ φ(f). These two conditions together say the bijection is not just object-by-object but globally coherent — it commutes with all morphisms in both categories. This is what makes an adjunction a structural relationship between the categories rather than a collection of independent accidents. Without naturality, the bijection at each (c, d) pair could be arbitrary and unrelated to bijections at other pairs.

Question 4

The hom-set bijection Hom_D(Lc, d) ≅ Hom_C(c, Rd) is merely a coincidence that holds separately for each pair of objects (c, d), with no coherence constraint relating bijections at different objects.

• T. True
• F. False

Answer: False

This is precisely what naturality rules out. An adjunction requires the bijection to be natural in both c and d — meaning it must commute with pre- and post-composition in a specific way. This naturality is the difference between an adjunction and a mere family of bijections. It ensures that the entire structure of both categories is respected: the bijection is not a coincidence at each point but a consequence of a global structural relationship between L and R. Without naturality, one could construct bijections Hom_D(Lc, d) ≅ Hom_C(c, Rd) at each (c, d) pair independently that do not constitute an adjunction.

Question 5

Why is naturality of the hom-set bijection φ: Hom_D(Lc, d) ≅ Hom_C(c, Rd) a stronger condition than saying it is a bijection for each pair (c, d)? What does naturality add?

Model answer: A bijection for each pair (c, d) independently could be completely arbitrary and unrelated across pairs — there would be no reason the bijection at (c, d) should be consistent with the bijection at (c', d) or (c, d'). Naturality adds coherence: the bijection must commute with all morphisms in both categories. Specifically, naturality in c requires that transposing and then precomposing gives the same result as precomposing and then transposing (via L). Naturality in d requires the same for postcomposition (via R). This means the transposition operation itself is a natural transformation, not just a collection of set bijections. It is this global coherence that makes adjunctions ubiquitous and structurally significant: free constructions, tensor products, products, and many other categorical constructions satisfy exactly this coherence condition, which is why adjunctions appear throughout mathematics.

A student who says 'naturality means the bijection is natural' is circular. The answer should articulate concretely what naturality adds: coherence with morphisms, or equivalently, that the bijection is functorial in both arguments. The contrast with a 'mere family of bijections' is essential to the answer.