A student studying 2-categories argues: 'Once I have defined vertical and horizontal composition and verified that each is independently associative, the interchange law must automatically hold.' Why is this reasoning incorrect?
AThe interchange law is automatic only for strict 2-categories; in bicategories it is replaced by the coherence theorem
BThe interchange law constrains how vertical and horizontal composition interact with each other — this interaction is not implied by each composition being individually associative; it is a separate, independent condition
CThe student is correct in Cat specifically, because the interchange law follows from naturality of composition there
DAssociativity is sufficient for the interchange law only when the 2-category has a single 0-cell
Associativity of vertical composition says (γ ∘ β) ∘ α = γ ∘ (β ∘ α) for 2-cells stacked along shared 1-cells. Associativity of horizontal composition says (γ ★ β) ★ α = γ ★ (β ★ α) for 2-cells arranged side by side. Neither says anything about mixing the two operations. The interchange law (β₂ ∘ β₁) ★ (α₂ ∘ α₁) = (β₂ ★ α₂) ∘ (β₁ ★ α₁) is an additional axiom that must be verified separately — it asserts that combining four 2-cells in a 2×2 grid gives the same result regardless of whether you compose rows first or columns first.
Question 2 Multiple Choice
In the 2-category Cat, which of the following is a 2-cell (2-morphism)?
AA small category C
BA functor F: C → D between two categories
CA natural transformation α: F ⇒ G between two functors with the same source and target categories
DAn adjunction L ⊣ R between two categories
In Cat, the three levels are: 0-cells are small categories, 1-cells are functors between categories, and 2-cells are natural transformations between functors with the same source and target. A natural transformation α: F ⇒ G assigns to each object X of C a morphism α_X: F(X) → G(X) in D, satisfying a naturality square. This is exactly the morphism-between-morphisms structure that defines 2-cells. Adjunctions involve more data than a single natural transformation and are not themselves 2-cells in this sense, though they can be characterized using 2-categorical structure.
Question 3 True / False
A 2-category is equivalent to a double category because both structures have objects, morphisms between objects, and morphisms between morphisms.
TTrue
FFalse
Answer: False
A 2-category has one kind of 1-cell (morphism between objects), and 2-cells are morphisms between those 1-cells. A double category has two distinct kinds of 1-cells — horizontal morphisms and vertical morphisms — which can compose independently, plus 2-cells that are squares with all four boundary types. The structures are genuinely different: Cat is naturally a 2-category, while the category of spans or of rings with bimodules is naturally a double category. The distinction matters for applications in topology, algebra, and theoretical computer science.
Question 4 True / False
In a strict 2-category, the interchange law must be explicitly imposed as an independent axiom, not derived from the associativity and unit laws for 1-cell and 2-cell composition.
TTrue
FFalse
Answer: True
The axioms of a strict 2-category include: (i) associativity and units for 1-cell composition, (ii) associativity and units for vertical composition of 2-cells (making each Hom(A, B) a category), and (iii) the interchange law relating vertical and horizontal composition. Each is independent. In Cat, the interchange law can be verified from the definition of horizontal composition (whiskering) and naturality — but this verification shows it holds in Cat, not that it follows from the abstract axioms. An abstract 2-category that violates the interchange law would be internally inconsistent.
Question 5 Short Answer
Explain in your own words what the interchange law requires and why it cannot be derived from the other axioms of a 2-category.
Think about your answer, then reveal below.
Model answer: The interchange law says that composing four 2-cells arranged in a 2×2 grid gives the same answer regardless of order: (β₂ ∘ β₁) ★ (α₂ ∘ α₁) = (β₂ ★ α₂) ∘ (β₁ ★ α₁). You can either compose rows first (two vertical composites) then compose those horizontally, or compose columns first (two horizontal composites) then compose those vertically — the result must agree. This is a coherence condition between two independent composition operations. It cannot be derived from the other axioms because those axioms govern each composition operation in isolation, saying nothing about how they interact when applied in succession.
The interchange law is analogous to the exchange rule in logic or the Eckmann-Hilton argument in higher algebra: when two binary operations interact on the same set of elements, their interaction must be constrained for the structure to be well-behaved. In Cat, you can verify the interchange law explicitly using the definition of whiskering and naturality squares. In an abstract 2-category, there is no such reduction — it must stand as an axiom. Violation would mean the order of applying vertical then horizontal (or vice versa) composition produces different 2-cells, making the structure incoherent.