Products, coproducts, initial and terminal objects

Builds on

Required prerequisites, all the way down: 17

You will learn to

• Identify initial and terminal objects in Set, Grp, rings and posets, or show that none exist.
• Compute products and coproducts in Set (disjoint union), Grp (free product), Ab (direct sum), Top and a poset (meet and join).
• Prove that products are unique up to unique isomorphism and that coproducts in C are products in Cᵒᵖ.

Definition

• An object T is terminal if every object has exactly one morphism to T; an initial object is defined dually.
• A product of A and B is an object A × B with morphisms π₁ : A × B → A and π₂ : A × B → B such that for every f : X → A and g : X → B there is a unique h : X → A × B with π₁ ∘ h = f and π₂ ∘ h = g.
• A coproduct in C is a product in Cᵒᵖ.

Theorem

• In Set, ∅ is initial, every one-element set is terminal, and coproducts are disjoint unions.
• In Grp the trivial group is both initial and terminal, and the coproduct of two groups is their free product.
• In a poset regarded as a category, products are greatest lower bounds and coproducts are least upper bounds, when they exist.
• In the category of rings with identity, ℤ is initial.
• A product of A and B is a terminal object in the category of cones over the pair (A, B).

Common misconceptions

• Assuming coproducts look like products: in Grp, ℤ ∗ ℤ is non-abelian, unlike ℤ × ℤ.
• Thinking every category has products.

Leads to

Optional discoveries

• The λ-calculus Product and sum types in programming languages correspond to products and coproducts.
• ← Products by maps: a first universal property

Further reading

• Tom Leinster, Basic Category Theory (2014)
• Emily Riehl, Category Theory in Context (2016)