The Yoneda lemma

Builds on

Required prerequisites, all the way down: 18

You will learn to

• State and prove the Yoneda lemma, including its naturality in A and F.
• Deduce that the Yoneda embedding C → [Cᵒᵖ, Set] is full and faithful, and that Hom(−, A) ≅ Hom(−, B) implies A ≅ B.
• Apply the Yoneda lemma in familiar settings, for example to recover Cayley's theorem for groups.

Definition

• For a locally small category C, the Yoneda embedding y : C → [Cᵒᵖ, Set] sends A to Hom(−, A).
• A presheaf on C is a functor Cᵒᵖ → Set.

Theorem

• Yoneda lemma: for every functor F : C → Set and object A, natural transformations Hom(A, −) ⇒ F correspond bijectively to elements of F(A), via α ↦ α_A(id_A).
• The Yoneda embedding is full and faithful; hence A ≅ B whenever Hom(−, A) ≅ Hom(−, B).
• Applied to a group regarded as a one-object category, the Yoneda lemma yields Cayley's theorem.
• Every presheaf on a small category is a colimit of representable presheaves.

Common misconceptions

• Reading Yoneda as saying an object 'is' its set of elements; it says an object is determined by the morphisms into it from all objects.
• Expecting the lemma to be hard to prove; its proof is short, and its power lies in how widely it applies.

Leads to

Optional discoveries

• ∞-categories An ∞-categorical Yoneda lemma also holds and is a basic tool of higher category theory.
• Homotopy type theory In homotopy type theory, path induction can be seen as a type-theoretic analogue of the Yoneda lemma for identity types.
• ← Products by maps: a first universal property
• ← Functors
• ← Adjunctions

Learning pathways

• Research preparation

Further reading

• Emily Riehl, Category Theory in Context (2016)
• Tom Leinster, Basic Category Theory (2014)
• Saunders Mac Lane, Categories for the Working Mathematician (1971)