In the category Ring of rings with ring homomorphisms, the inclusion ι: ℤ → ℚ is an epimorphism. Why, even though ℚ contains elements (non-integer rationals) that ι never reaches?
ABecause ι is surjective onto a dense subset of ℚ, and dense maps are always epimorphisms
BBecause any ring homomorphism f: ℚ → R is completely determined by its value on ℤ, so if g ∘ ι = h ∘ ι then g = h everywhere on ℚ
CBecause ℤ and ℚ are isomorphic as abelian groups, making the inclusion an isomorphism
DBecause epimorphism in Ring just means the map has dense image, which ι has
Epimorphism means right-cancellability: g ∘ ι = h ∘ ι implies g = h. For ring homomorphisms out of ℚ, if g and h agree on all integers, they must agree on all rationals: f(p/q) = f(p) · f(q)^{−1} is forced by the homomorphism axioms, and f(p) is determined by f(1). So any two ring homomorphisms ℚ → R that agree on ℤ must be identical — ι is right-cancellable. This shows epimorphism is NOT about surjectivity; it's about whether morphisms out of the codomain are uniquely determined by precomposition. Dense image (option D) is the correct characterization in Top (topological spaces), not Ring.
Question 2 Multiple Choice
A morphism f: A → B in a category is a monomorphism if and only if it has a left inverse (a morphism g: B → A with g ∘ f = id_A).
ATrue — only morphisms with left inverses satisfy the left-cancellability condition
BFalse — having a left inverse (being a split monomorphism) is sufficient for being monic, but not necessary; there are monomorphisms without left inverses
CTrue — in any category, monic and split monic are equivalent concepts
DFalse — left inverses make morphisms epic, not monic
Having a left inverse implies being monic (split monics are monic), but the converse fails. In Ab, the inclusion ℤ → ℚ is monic (injective) but has no left inverse as an abelian group homomorphism — any homomorphism ℚ → ℤ must send 1/n to an element r with n·r = 1 in ℤ, which is impossible for n > 1. The monomorphism condition (left-cancellability: f ∘ g = f ∘ h ⟹ g = h) is strictly weaker than having a left inverse. Confusing the two is a common error when moving from Set (where all injections split) to general categories.
Question 3 True / False
In the category Set, every epimorphism is surjective.
TTrue
FFalse
Answer: True
In Set, epimorphisms coincide exactly with surjective functions. If f: A → B is not surjective, there exists an element b ∈ B not in the image of f. Define two functions g, h: B → {0,1} where g is the constant 0 function and h sends b to 1 and everything else to 0. Then g ∘ f = h ∘ f (both are constant 0 on A), but g ≠ h — so f is not right-cancellable, hence not epic. This makes Set a well-behaved category where the categorical definition aligns perfectly with the set-theoretic one.
Question 4 True / False
In the category of topological spaces (Top) with continuous maps, the epimorphisms are exactly the surjective continuous maps.
TTrue
FFalse
Answer: False
In Top, the epimorphisms are the *dense* continuous maps — maps whose image is dense in the codomain (every open set intersects the image) — not just the surjective ones. A surjection is always dense (and hence epic), but a dense map need not be surjective. This is the same divergence as ℤ → ℚ in Ring: the codomain is 'generated' in the appropriate sense by the image, even though the image doesn't cover every point. This example illustrates why 'epic ≠ surjective' is a structural feature of many natural categories.
Question 5 Short Answer
Explain why category theory defines monomorphisms via left-cancellability rather than via 'no two inputs give the same output,' and what is gained by the categorical definition.
Think about your answer, then reveal below.
Model answer: The element-based definition ('injective: f(a) = f(b) ⟹ a = b') requires objects to have elements, which is not the case in an arbitrary category. Left-cancellability (f ∘ g = f ∘ h ⟹ g = h) is expressed entirely in terms of morphisms and composition, requiring no reference to elements. The gain is universality: the categorical definition applies to categories of sets, groups, rings, topological spaces, vector spaces, and purely abstract categories — and it reveals which properties of injectivity are truly 'structural' versus which depend on set-theoretic membership. The divergence between mono and injective in some categories (and their coincidence in others) then becomes a theorem that tells you something about the category itself.
This is the core move of category theory: reformulate element-dependent definitions in terms of morphisms. The result is a definition that is simultaneously more abstract and more powerful — it generalizes to settings where 'element' has no meaning and, in concrete categories, recovers the familiar notion while revealing its true algebraic content.