Questions: Reflective and Coreflective Subcategories

Question 1

A student says: 'The category Ab of abelian groups is reflective inside Grp because we can quotient any group G by its commutator subgroup [G,G] to get an abelian group. That's the whole story.' What crucial fact is missing from this description?

• A. The construction is wrong — the reflection is the center Z(G), not the commutator quotient
• B. The construction is not functorial and cannot be applied consistently to group homomorphisms
• C. The reflection G → G/[G,G] has a universal property: every group homomorphism from G to any abelian group factors uniquely through it — this is what makes it the *reflector*, not just any quotient
• D. Ab is not reflective in Grp; it is coreflective because the inclusion has a right adjoint

Answer: (C)

The abelianization construction G ↦ G/[G,G] is correct, but the student describes it as if the quotient is just a convenient construction. The key is the *universal property*: G/[G,G] is the initial abelian group that G maps to. Any group homomorphism f: G → A with A abelian factors uniquely as G → G/[G,G] → A. This universal factorization property is what makes it a reflector — the reflection is not merely *an* abelian quotient but the *best* one. Without the universal property, you just have a quotient; with it, you have an adjunction.

Question 2

Which of the following correctly characterizes the reflector L: C → D in a reflective subcategory D ⊆ C?

• A. L is a functor that sends each object of C to an arbitrarily chosen object of D
• B. For each object X in C, L(X) is the object of D such that any morphism from X to any object A in D factors uniquely through the unit map η_X: X → L(X) in C
• C. L is the right adjoint of the inclusion functor i: D ↪ C
• D. L is surjective on objects: every object of D appears as L(X) for some X in C

Answer: (B)

The reflector L is characterized by the universal property of the unit map η_X: X → L(X). This map is not just any morphism into D — it is the *initial* morphism from X to any D-object, meaning all other maps from X into D factor through it uniquely. This is the adjunction condition Hom_D(L(X), A) ≅ Hom_C(X, i(A)) made concrete. Option C has the variance wrong: L is the *left* adjoint of the inclusion, not the right adjoint. Being a left adjoint (the reflector) is what defines a reflective subcategory.

Question 3

The Stone-Čech compactification βX is the reflection of a topological space X into the subcategory of compact Hausdorff spaces, meaning every continuous map from X to any compact Hausdorff space factors uniquely through the canonical map X → βX.

• T. True
• F. False

Answer: True

This is precisely the universal property of the Stone-Čech compactification, and it is exactly the statement that compact Hausdorff spaces form a reflective subcategory of completely regular spaces with βX as the reflector. The canonical map X → βX is the unit of the adjunction. Any continuous f: X → K with K compact Hausdorff extends uniquely to a continuous f̃: βX → K. This universal property is what distinguishes βX from all other compactifications of X — it is the *maximal* compactification in the sense that it maps onto all others.

Question 4

Nearly every full subcategory of a category is reflective, as long as it is closed under isomorphisms.

• T. True
• F. False

Answer: False

Being a full subcategory closed under isomorphisms is not sufficient for reflectivity. Reflectivity requires that the inclusion functor i: D ↪ C has a left adjoint — a reflector L: C → D — satisfying a universal property for every object of C. Many full subcategories do not have this property. For example, the subcategory of finite sets inside all sets is a full subcategory closed under isomorphisms, but it is not reflective: there is no 'best finite approximation' to an infinite set in the required sense. The existence of the adjoint is a genuine and nontrivial condition.

Question 5

What does it mean for a subcategory D to be reflective in C, and why is the universal property of the unit map η_X: X → L(X) the central fact — rather than merely the existence of a functor L: C → D?

Model answer: D is reflective in C if the inclusion functor i: D ↪ C has a left adjoint L: C → D. This means for every X ∈ C there is a morphism η_X: X → L(X) in C such that any morphism f: X → A with A ∈ D factors uniquely as f = g ∘ η_X for some g: L(X) → A in D. The universal property is central because it is what makes L(X) the *best* D-approximation to X, not just *some* D-object related to X. Without it, L is just a functor that maps into D — many such functors exist and most have no special status. The universal property gives L(X) a canonical role: it is the initial object in the category of morphisms from X to D-objects, which is precisely the adjunction condition.

The distinction matters practically: sheafification is the unique functor from presheaves to sheaves that is left adjoint to the inclusion, and this determines it up to unique isomorphism. Without the universal property, you would have no canonical way to extend maps from presheaves to sheaves, and the construction would lose its functorial coherence. The universal property is not extra structure on top of the functor — it *is* the reason the reflector is well-defined and useful.