Questions: The General Adjoint Functor Theorem

Question 1

A functor G: D → C is known to preserve all small limits. Can you conclude that G has a left adjoint?

• A. Yes — limit preservation is sufficient for left adjoint existence by the General Adjoint Functor Theorem
• B. Yes — any limit-preserving functor between complete categories is automatically a right adjoint
• C. No — limit preservation is necessary but not sufficient; the solution set condition must also hold
• D. No — G would need to also preserve all colimits to guarantee a left adjoint exists

Answer: (C)

Limit preservation is necessary (right adjoints always preserve limits) but not sufficient. The General Adjoint Functor Theorem requires an additional condition — the solution set condition — which controls the set-theoretic size of the 'search space' for the universal arrow that defines the left adjoint. Without it, the collection of candidate solutions might form a proper class, preventing the limit-based construction from working. The misconception that limit preservation alone suffices is common and worth explicitly refuting.

Question 2

The General Adjoint Functor Theorem guarantees that the forgetful functor G: Grp → Set has a left adjoint. What is that left adjoint?

• A. The free group functor F: Set → Grp, which constructs the free group on any given set of generators
• B. The quotient group functor, which maps sets to groups by collapsing equivalence classes
• C. The powerset functor, which maps each set to the group of all its subsets under symmetric difference
• D. The opposite group functor, which reverses all group operations on a set

Answer: (A)

The free group functor is the canonical left adjoint to the forgetful functor from Grp to Set. The adjunction says: group homomorphisms from F(S) to G are in natural bijection with set functions from S to |G|. The GAFT guarantees this left adjoint exists because the forgetful functor preserves limits (a product of groups forgets to a product of sets) and the solution set condition holds (generators for any group can be taken from the underlying set). The theorem proves existence; the explicit construction of free groups provides the description.

Question 3

The General Adjoint Functor Theorem provides an explicit formula for constructing the left adjoint of G once its existence has been very likely by the theorem's conditions.

• T. True
• F. False

Answer: False

The theorem guarantees existence — it proves that a left adjoint must exist — but the construction it provides (taking a limit over the solution set) is often not a useful explicit description. In practice, you typically need additional work to identify what the left adjoint actually does on objects and morphisms. For example, the theorem tells you the free group functor must exist, but you still need the explicit construction (equivalence classes of words) to work with it concretely. Existence and explicitness are separate concerns.

Question 4

If a functor G fails to preserve even one small limit, then G cannot be a right adjoint to any functor — no matter what other properties G might have.

• T. True
• F. False

Answer: True

This follows from a fundamental theorem of adjunctions: right adjoints always preserve limits. The proof uses the unit-counit characterization of adjunctions — the natural bijection Hom(FC, D) ≅ Hom(C, GD) forces G to commute with limits. So limit preservation is a necessary condition, not just a sufficient one. If G fails to preserve a specific limit, you have a certificate of non-adjointness — you can immediately conclude G is not a right adjoint to anything.

Question 5

What does the solution set condition say, and why is it needed in the General Adjoint Functor Theorem in addition to the requirement that G preserves limits?

Model answer: For each object C in C, consider all pairs (D, f) where f: C → G(D) is a morphism into the image of G. For a left adjoint F(C) to exist, there must be a universal such morphism — the unit η_C: C → G(F(C)). The solution set condition says: there exists a set {D_i, f_i} of such pairs through which every other morphism C → G(D) factors. This set controls the 'size' of the search: once you have a solution set, you can form a limit over it to construct the universal morphism F(C). Without this condition, the candidate solutions could form a proper class, and the limit construction would not be valid in standard set theory.

The solution set condition is the technical bridge between 'G preserves limits' (necessary) and 'G has a left adjoint' (conclusion). It prevents set-theoretic pathology by bounding the problem to a set. In many practical settings — locally small categories, accessible functors — the condition holds automatically, which is why the theorem is widely applicable without explicitly verifying the solution set in each case.