The forgetful functor U: Grp → Set sends each group to its underlying set. It is a right adjoint to the free group functor F: Set → Grp. Does U preserve products — i.e., is U(G₁ × G₂) ≅ U(G₁) × U(G₂)?
ANot necessarily — U forgets group structure, so it might not respect the product construction
BYes — right adjoints preserve all limits, and products are limits, so U must preserve products
CYes, but only for abelian groups, where products and coproducts coincide
DWhether U preserves products depends on the specific groups G₁ and G₂, not on U being a right adjoint
Preservation of limits is the key structural property of ALL right adjoints, not just some. Since U is a right adjoint, it preserves every limit that exists in Grp — including products, terminal objects, equalizers, and pullbacks. Concretely, the underlying set of G₁ × G₂ is indeed the cartesian product of the underlying sets, which is exactly what the theorem predicts. The 'forgetting structure' intuition is misleading — forgetful functors are often right adjoints precisely because they represent the 'remembering the structure is there' view.
Question 2 Multiple Choice
For an adjunction L ⊣ R with counit ε_d: L(Rd) → d, what does the counit represent in terms of the adjunction's universal property?
AThe identity morphism on d, showing R and L are inverse equivalences
BThe universal morphism expressing that any map Lc → d factors uniquely through L(Rd) → d via the corresponding map c → Rd in C
CThe unit η of the adjunction applied to the object Rd
DA natural isomorphism between L and R showing they define an equivalence of categories
The counit ε_d: L(Rd) → d is the 'evaluation' morphism for the right adjoint. By the hom-bijection, a map c → Rd in C corresponds to a map Lc → d in D. The counit is the specific morphism corresponding to the identity on Rd — it is the universal such map, in the sense that every Lc → d factors as Lc → L(Rd) → d where the first map comes from the unique c → Rd and the second is ε_d. This is the precise sense in which d is a 'universal target' for maps from the image of L.
Question 3 True / False
If R: D → C is a right adjoint and D has products, then R preserves products: R(d₁ × d₂) ≅ R(d₁) × R(d₂) in C.
TTrue
FFalse
Answer: True
Products are limits (they satisfy a universal cone property), and right adjoints preserve all limits. This is the single most important structural theorem about right adjoints. The proof follows from the limit-preservation property, which is itself a consequence of the hom-bijection: the universal property of limits in D transfers through the natural bijection to establish the universal property of the image under R in C.
Question 4 True / False
To show that a right adjoint R preserves limits, one should verify each type of limit separately — products, equalizers, terminal objects, and pullbacks each require a distinct argument.
TTrue
FFalse
Answer: False
The theorem that right adjoints preserve limits is a single, uniform result. Once you know R is a right adjoint, all limit preservation follows immediately — products, equalizers, terminal objects, pullbacks, and every other limit shape. No separate verification is needed for each type. This is the power of the categorical framework: a single structural fact (R is a right adjoint) implies an entire family of preservation results simultaneously.
Question 5 Short Answer
Explain why right adjoints preserve limits. Use the duality between left and right adjoints to make the argument as concise as possible.
Think about your answer, then reveal below.
Model answer: A functor R is a right adjoint iff its opposite R^op: D^op → C^op is a left adjoint. Left adjoints preserve colimits. Limits in D correspond to colimits in D^op, and limits in C correspond to colimits in C^op. Since R^op (a left adjoint) sends colimits in D^op to colimits in C^op, passing back through the opposite-category correspondence shows R sends limits in D to limits in C. Right adjoint preserves limits by duality from the fact that left adjoints preserve colimits.
This duality argument is the cleanest proof available — it converts the right-adjoint case into the left-adjoint case for free, requiring no additional work. The slogan is: 'Left adjoints preserve colimits, right adjoints preserve limits' — and these two facts are not independent theorems but dual statements about the same underlying structure viewed from opposite categories.