Questions: Left Adjoint Functors

Question 1

A student is asked whether the forgetful functor U: Grp → Set (which sends a group to its underlying set) has a left adjoint. The student reasons: 'U just forgets structure — it doesn't construct anything new — so it can't have a left adjoint.' What is wrong with this reasoning?

• A. The student is correct; forgetful functors are right adjoints, never left adjoints
• B. The student conflates the direction of the adjunction: U has a left adjoint F: Set → Grp, the free group functor, where group homomorphisms F(S) → G correspond bijectively to functions S → U(G)
• C. The forgetful functor U does not actually exist because you cannot ignore group structure in a category-theoretic setting
• D. The student should check whether U preserves limits, not whether it has a left adjoint

Answer: (B)

The adjunction F ⊣ U means Hom_Grp(F(S), G) ≅ Hom_Set(S, U(G)): a group homomorphism out of the free group on S corresponds exactly to a function from the generating set S to the underlying set of G. The forgetful functor U is the *right* adjoint here — it is F (the free group functor) that is the left adjoint. The student's error is thinking that 'not constructing anything' means 'no adjoint'; in fact, forgetful functors frequently have left adjoints (free constructions) precisely because the bijection Hom(Lc, d) ≅ Hom(c, Rd) trades structured morphisms for unstructured ones, which is exactly what forgetfulness enables.

Question 2

The free group functor F: Set → Grp is a left adjoint and therefore preserves colimits. What does this mean concretely when applied to a disjoint union of sets?

• A. The free group on any set has no non-trivial subgroups
• B. The free group on a disjoint union S ⊔ T is isomorphic to the free product F(S) * F(T) — the coproduct of the free groups
• C. The free group functor maps every colimit to a limit
• D. F preserves the number of elements: |F(S)| equals the cardinality of the colimit diagram

Answer: (B)

Colimits in Set include disjoint unions (coproducts). Left adjoints preserve colimits, so F(S ⊔ T) ≅ F(S) ⊔_Grp F(T). The coproduct in Grp is the free product *, so F(S ⊔ T) ≅ F(S) * F(T). Concretely: the free group on a set formed by combining two disjoint generating sets is generated by all elements of both sets with no relations between them — precisely the free product. This is an elegant instance of the general theorem: to compute a left adjoint on a colimit, compute it on each piece and take the colimit of the results.

Question 3

In an adjunction L ⊣ R, a single morphism f: Lc → d in D determines a unique corresponding morphism φ(f): c → Rd in C, and knowing either side completely determines the other.

• T. True
• F. False

Answer: True

The bijection φ_{c,d}: Hom_D(Lc, d) → Hom_C(c, Rd) is exactly that — a bijection. Every morphism on one side corresponds to exactly one morphism on the other, with no ambiguity. This one-to-one correspondence is what makes the adjunction a precise mathematical relationship rather than a loose analogy. In the free group case: every group homomorphism F(S) → G is completely determined by a unique function S → U(G), and every such function extends to a unique homomorphism. The naturality condition further ensures this bijection respects composition and is not just a set-theoretic coincidence.

Question 4

Left adjoint functors preserve limits (products, pullbacks, equalizers) because they interact with the structure of both categories in the adjunction.

• T. True
• F. False

Answer: False

This reverses the fundamental theorem about adjoints. Left adjoints preserve *colimits* (coproducts, pushouts, coequalizers, initial objects). Right adjoints preserve *limits* (products, pullbacks, equalizers, terminal objects). This is one of the most important and frequently confused facts in category theory. The proof uses the hom-set bijection: Hom(L(colim F), d) ≅ Hom(colim F, Rd) ≅ lim Hom(F(−), Rd) ≅ lim Hom(L(F(−)), d), showing L(colim F) satisfies the universal property of colim(L∘F). No analogous argument works for limits of left adjoints.

Question 5

What does it mean for the bijection Hom_D(Lc, d) ≅ Hom_C(c, Rd) to be 'natural in both c and d,' and why does naturality matter?

Model answer: Naturality means the bijection commutes with pre- and post-composition by morphisms in C and D. Formally, for any morphism g: c' → c in C, the bijection maps f ∘ L(g) (precomposing on the Lc side) to φ(f) ∘ g (precomposing on the c side); and for any h: d → d' in D, it maps h ∘ f to R(h) ∘ φ(f). These coherence conditions mean the bijection is not an arbitrary set-theoretic correspondence but respects all categorical structure. Without naturality, two categories could have matching hom-sets by coincidence; with naturality, the bijection is a structural relationship that propagates coherently through the entire categorical framework.

Naturality is what elevates the hom-set bijection from a curiosity to a powerful tool. It means you can use the adjunction to transport information about morphisms along any path in the category, and the bijection will remain consistent. In practice, it means: to compute what L does on a composite morphism, you can either compute the composite first and then apply L, or compute L on each piece — and the adjunction bijection gives the same answer either way. This coherence is why adjunctions interact so well with limits, colimits, and natural transformations, making them central to essentially all of modern pure mathematics.