A square diagram has objects A, B, C, D with morphisms f: A → B, g: B → D, h: A → C, k: C → D. The diagram is said to commute. What exactly does this assert?
AAll four morphisms f, g, h, k are equal to each other as arrows in the category
BThe composite g∘f equals the composite k∘h — the two paths from A to D yield the same result
CEvery morphism in the diagram factors through every other object, meaning f = h and g = k
DThe diagram has exactly one morphism between each pair of objects, so there is only one path and commutativity is automatic
Commutativity asserts that specific paths between the same source and target produce equal composites. In this square, the two paths from A to D are: A→B→D (composite g∘f) and A→C→D (composite k∘h). The diagram commutes if and only if g∘f = k∘h. The morphisms f, g, h, k are generally distinct arrows — for example, f: A→B and h: A→C have different codomains, so they cannot possibly be equal. Commutativity is not a claim about individual morphisms being equal; it is a claim about equality of composed paths between designated endpoints.
Question 2 Multiple Choice
In a commutative triangle with morphisms f: A→B, g: B→C, and h: A→C, what equation does commutativity assert, and what type of object is on each side?
Af = g∘h, asserting that the direct morphism equals a composite; both sides are morphisms A→B
Bh = g∘f, asserting the direct morphism A→C equals the composite going through B; both sides are morphisms A→C
Cg = h∘f, asserting the morphism B→C equals a composite; both sides are morphisms B→C
Df∘g = h, asserting that the order of composition determines commutativity; both sides are endomorphisms
In the triangle, the two paths from A to C are: directly via h, and through B via f then g (written g∘f in standard notation where composition is applied right-to-left). Commutativity asserts h = g∘f. Both sides are morphisms A→C — they have the same source and target, which is required for the equality to even be well-typed. Note option A reverses f and h and gets the composition order wrong; option C confuses which morphism is claimed equal to the composite.
Question 3 True / False
A commutative diagram asserts that most morphisms appearing in the diagram are equal to one another, since most paths lead to the same objects.
TTrue
FFalse
Answer: False
This is the most common misreading of commutative diagrams. Commutativity asserts that certain *composite paths* with the same source and target are equal — not that the individual morphisms are equal. In a commutative square, g∘f = k∘h, but f, g, h, and k are generally distinct arrows. Indeed, f: A→B and h: A→C cannot even be equal (they have different codomains). The diagram encodes a specific equality between two composite paths, nothing more. Different commutative diagrams in the same category encode different equalities — the diagram is the proof obligation, not a claim about all morphisms.
Question 4 True / False
When a definition in category theory states 'the following diagram commutes,' it is making a constructive assertion: it is claiming that specific morphisms are chosen or constructed so that the stated path equalities hold by design.
TTrue
FFalse
Answer: True
This is a crucial distinction between two uses of commutative diagrams. When a diagram appears in a *theorem*, commutativity is something to be *proved* — it is a consequence of other axioms or hypotheses. When a diagram appears in a *definition* (such as the definition of a natural transformation or a pullback), commutativity is *imposed by construction* — the objects and morphisms being defined are required to make the diagram commute, and this requirement is what gives them their universal property. Reading 'the following diagram commutes' in a definition tells you what equalities must hold for something to count as an instance of that defined concept.
Question 5 Short Answer
Two students examine a square commutative diagram showing paths A→B→D and A→C→D. One student says: 'So the morphism f: A→B must equal the morphism h: A→C, since they both start at A.' Explain what's wrong with this reasoning and state what the diagram actually asserts.
Think about your answer, then reveal below.
Model answer: The student's error is conflating the morphisms themselves with the paths they participate in. The morphism f: A→B and the morphism h: A→C have different codomains (B vs. C), so they cannot be equal — they live in different hom-sets. Commutativity makes no claim about individual morphisms being equal. What the diagram asserts is that the two full paths from A to D produce the same composite: g∘f = k∘h (where g: B→D and k: C→D). Both sides are morphisms A→D — same source, same target — so the equality is well-typed. The claim is about the result of traversing each complete path, not about any individual step.
This error is common because students borrow geometric intuition (all roads lead to the same destination, so the roads must be the same) and apply it to categorical composition. In geometry, two paths from A to D that are equal in length might seem 'the same,' but in category theory, morphisms are abstract arrows with sources and targets — what matters is the equality of their composites, not the equality of the arrows themselves. Keeping the type-checking in mind (checking that source and target match before asserting equality) prevents this confusion.