Questions: Commutative Diagrams in Category Theory

Question 1

A category has objects A, B, C, D and morphisms f: A → B, g: B → D, h: A → C, k: C → D. It is known that g ∘ f ≅ k ∘ h — the two compositions are isomorphic as objects of a functor category, but not equal as morphisms. Does the square commute?

• A. Yes, because the two paths yield isomorphic objects, which is sufficient for commutativity in any reasonable category
• B. No, commutativity of a diagram requires the two compositions to be equal as morphisms in the hom-set, not merely isomorphic
• C. Yes, because isomorphic morphisms are equal in any skeletal category, and every category is equivalent to a skeletal one
• D. No, but the diagram is still considered homotopy-commutative, which is sufficient for most categorical purposes

Answer: (B)

Commutativity is a strict equality condition: g ∘ f = k ∘ h must hold as morphisms in the hom-set Hom(A, D). Isomorphism is weaker — it says there exists an invertible morphism between the two results, but does not say the morphisms themselves are the same arrow. Relaxing to isomorphism requires specifying the isomorphism, tracking coherence conditions between multiple such isomorphisms, and verifying compatibility — this is the subject of higher category theory (2-categories, ∞-categories). Classical commutative diagram arguments depend on strict equality; the power of the notation comes precisely from this strictness.

Question 2

What does it mean for a functor F: C → D to 'preserve composition'?

• A. F assigns to each object of C an object of D, and to each morphism a morphism of the same type
• B. For every morphism f in C, F(f) has the same domain and codomain as f in the same category
• C. Every commutative diagram in C is sent to a commutative diagram in D — that is, F(g ∘ f) = F(g) ∘ F(f)
• D. F is injective on hom-sets, so distinct morphisms in C map to distinct morphisms in D

Answer: (C)

Functoriality requires two things: F(id_A) = id_{F(A)} (identity preservation) and F(g ∘ f) = F(g) ∘ F(f) (composition preservation). The composition law says exactly that if a triangle or square commutes in C — if two paths compose to the same morphism — then their images under F also compose to the same morphism in D. In diagram language: every commutative diagram in C maps to a commutative diagram in D. This is the precise meaning of 'F respects the structure of C.'

Question 3

A diagram drawn with objects as vertices and morphisms as arrows automatically commutes, because drawing it implies the depicted relationships are satisfied.

• T. True
• F. False

Answer: False

This is one of the most common misconceptions in category theory. Drawing a diagram only requires that morphisms exist with the depicted domains and codomains — it makes no assertion about commutativity. Commutativity is an additional equality condition on the compositions, which must be separately proven or explicitly stipulated. An author might draw a diagram to illustrate the objects and morphisms involved in a construction, then separately state 'and this diagram commutes' as a theorem to be proven. If diagrams automatically commuted by virtue of being drawn, there would be nothing to prove.

Question 4

The naturality condition for a natural transformation η: F ⇒ G requires, for every morphism f: A → B in the source category, that a specific square involving η_A, η_B, F(f), and G(f) commutes.

• T. True
• F. False

Answer: True

The naturality square for f: A → B states: η_B ∘ F(f) = G(f) ∘ η_A. This must hold for every morphism f in the source category C, not just for objects. The square has corners F(A), F(B), G(A), G(B), with F(f) and G(f) as horizontal edges and η_A, η_B as vertical edges. A natural transformation is not just a family of morphisms indexed by objects — it is a family of morphisms that makes all these squares commute. Commutativity of the naturality squares is exactly what 'natural' means.

Question 5

Explain why the distinction between 'isomorphic' and 'equal' is crucial for the definition of commutativity in a diagram.

Model answer: Commutativity requires that two different paths of composition yield the same morphism — exact equality in the hom-set, not merely an isomorphism between the results. If the condition were relaxed to isomorphism, we would need to specify the isomorphism itself (adding extra data), verify that multiple such isomorphisms cohere with each other (coherence conditions), and ensure these isomorphisms are compatible with composition (further axioms). The strictness of equality is what makes commutative diagrams a clean, self-contained notation: you simply verify that composing morphisms along different paths gives the identical arrow, with no auxiliary data, no coherence obligations, and no additional proof burden.

Higher category theory (2-categories, ∞-categories) does relax commutativity to 'commutes up to a specified isomorphism,' which is the correct setting for homotopy theory and derived algebraic geometry. But this requires a substantially richer framework. Classical category theory works with strict equality, and the power of commutative diagram arguments — their transferability between algebra, topology, and logic — rests on this strictness.