You know that the tensor product functor M ⊗ − is left adjoint to Hom(M, −). A colleague claims that therefore M ⊗ − preserves all limits — products, kernels, equalizers. What is wrong with this reasoning?
ANothing — left adjoints preserve all limits by the adjoint functor theorem
BThis is backwards: it is right adjoints that preserve limits; left adjoints preserve colimits (coproducts, cokernels, pushouts)
CTensor product preserves limits only when M is a flat module, so the claim holds only in special cases
DHom(M, −) preserves limits and since it is the adjoint of M ⊗ −, they must preserve the same limits
The rule is: left adjoints preserve colimits, right adjoints preserve limits. M ⊗ − sits on the left, so it preserves coproducts, cokernels, and filtered colimits — but not kernels or products in general. Hom(M, −) sits on the right, so it preserves kernels and products — but not cokernels. This is why M ⊗ − is right-exact (preserves cokernels as a left adjoint colimit-preservation) but not left-exact (fails to preserve kernels).
Question 2 Multiple Choice
The free-forgetful adjunction F ⊣ U between Set and Grp illustrates the natural isomorphism Hom_Grp(F(S), G) ≅ Hom_Set(S, U(G)). What does this say about free groups?
AEvery group homomorphism between free groups is determined solely by the cardinalities of their generating sets
BThere is a natural bijection: group homomorphisms from the free group F(S) to any group G correspond exactly to functions from the set S into the underlying set U(G) — this is the universal property of free groups, stated categorically
CFree groups are the 'smallest' groups, which is why they appear on the left side of the category arrow
DThe forgetful functor U is the categorical inverse of F, so composing them recovers the original set or group exactly
The adjunction isomorphism says: to define a group homomorphism out of a free group, you only need to specify where the generators go — any function from the generating set into the target group extends uniquely to a homomorphism. This is the universal property of free groups, now expressed as an adjunction. Option D is the classic misconception: adjoints are not inverses. U(F(S)) is the underlying set of the free group on S, which contains infinitely many words — not just S.
Question 3 True / False
If F ⊣ G (F is left adjoint to G), then F and G are inverse functors: applying F then G, or G then F, returns the original object unchanged.
TTrue
FFalse
Answer: False
Adjoint functors are not inverses. Inverse functors (equivalences of categories) satisfy F ∘ G ≅ id and G ∘ F ≅ id. Adjoint functors satisfy a much weaker condition: a natural isomorphism of hom-sets Hom_D(F(−), −) ≅ Hom_C(−, G(−)). For the free-forgetful pair, U(F(S)) is the underlying set of the free group on S — an infinite set of words, not the original finite set S. F and U are far from inverses; the relationship is structural, not invertible.
Question 4 True / False
The existence of derived functors Tor₁(M, −) and Ext¹(M, −) is a direct consequence of M ⊗ − and Hom(M, −) failing to preserve certain limits or colimits that their left/right adjoint status predicts they 'should' handle better.
TTrue
FFalse
Answer: True
Left adjoints preserve all colimits; right adjoints preserve all limits. M ⊗ − (left adjoint) preserves colimits but fails to preserve some limits — specifically kernels. This failure (the failure of M ⊗ − to be left-exact) is measured by Tor₁(M, −). Hom(M, −) (right adjoint) preserves limits but fails to preserve some colimits — specifically cokernels. This failure is measured by Ext¹(M, −). Derived functors are exactly the algebraic measurement of adjoint failure; they arise precisely where an adjoint cannot do what its position predicts.
Question 5 Short Answer
Why does a functor's position as 'left' versus 'right' adjoint determine what it preserves? Give the rule and one concrete consequence.
Think about your answer, then reveal below.
Model answer: Left adjoints preserve colimits (coproducts, pushouts, coequalizers, filtered colimits); right adjoints preserve limits (products, pullbacks, equalizers). This follows from the universal property structure of adjunctions and can be proved once, then applied universally. A concrete consequence: M ⊗ − is left adjoint, so it distributes over direct sums (a colimit: M ⊗ (A ⊕ B) ≅ (M ⊗ A) ⊕ (M ⊗ B)) but need not preserve kernels. This is why tensoring is right-exact but not necessarily left-exact, and why projective modules (over which tensoring is exact) are special.
The limit/colimit preservation theorem is one of the most powerful tools in category theory because it turns a question ('what does this functor preserve?') into a structural lookup ('which side of the adjunction is it on?'). It unifies dozens of separate algebraic facts — distributivity of tensor over direct sums, left-exactness of Hom, exactness of free modules — into a single categorical principle.