Questions: Products and Coproducts

Question 1

What is the coproduct of two sets A and B in the category Set?

• A. The Cartesian product A × B
• B. The disjoint union A ⊔ B
• C. The intersection A ∩ B
• D. The set of all functions from A to B

Answer: (B)

The coproduct in Set is the disjoint union A ⊔ B. It is equipped with injections i₁: A → A ⊔ B and i₂: B → A ⊔ B, and for any set C with functions f: A → C and g: B → C there is a unique function [f,g]: A ⊔ B → C satisfying [f,g]∘i₁ = f and [f,g]∘i₂ = g. The Cartesian product A × B is the categorical product, not the coproduct — a common confusion.

Question 2

In most category, the product A × B and the coproduct A + B of two objects are isomorphic to each other.

• T. True
• F. False

Answer: False

This is only true in special categories. In Ab (abelian groups), finite products and coproducts coincide as the direct sum A ⊕ B — this is a special property of abelian categories called a 'biproduct.' But in Set, A × B (Cartesian product) and A ⊔ B (disjoint union) have different cardinalities and are generally non-isomorphic. In Grp, the product is the direct product while the coproduct is the free product — very different structures.

Question 3

The universal property of the categorical product requires that the pairing morphism ⟨f,g⟩ be unique. Why is uniqueness essential — what would fail if two distinct morphisms both commuted with the projections?

Model answer: If two distinct morphisms h, k: C → A×B both satisfied π₁∘h = f, π₂∘h = g and π₁∘k = f, π₂∘k = g, then A×B would not be a limit — it would merely be an object with projections. Uniqueness forces the product to be the 'most economical' or 'universal' object satisfying the factoring condition, ensuring that any two constructions satisfying the universal property are uniquely isomorphic. Without uniqueness, the universal property cannot determine the product up to unique isomorphism.

Uniqueness is the defining feature that distinguishes universal properties from weaker existence conditions. It guarantees that different constructions of the product (e.g., ordered pairs vs. tagged unions) are not just isomorphic but canonically so, which is why category theory can work 'up to isomorphism' without losing information.