Yoneda

Yoneda

Yoneda refers to results in category theory that connect objects of a category with the way they represent or “pointwise” interact with other objects. The central results are the Yoneda lemma and the Yoneda embedding, both due to Nobuo Yoneda in the 1950s.

Yoneda lemma

Let C be a locally small category. For any object A in C and any presheaf F: C^op → Set, there is a natural bijection between the set of natural transformations from Hom(-, A) to F and the set F(A). In symbols: Nat(Hom(-, A), F) ≅ F(A). There is a dual form for covariant functors: Nat(Hom(A, -), F) ≅ F(A) for F: C → Set. The bijection is natural in both A and F. The lemma provides a precise way to recover the value of a presheaf at A from the way A represents morphisms into F.

Yoneda embedding

The Yoneda embedding y: C → [C^op, Set] sends each object A to the representable functor y(A) = Hom(-, A). This embedding is fully faithful, meaning that C can be viewed concretely inside the category of presheaves on C. Representable functors y(A) encode all morphisms into A, and the canonical identifications given by the lemma allow one to translate problems about C into problems about presheaves.

Consequences and uses

The Yoneda lemma provides a powerful tool for reasoning about objects via their morphisms, allowing the reconstruction of objects from their relationships with all other objects. It underpins the definition and study of limits, colimits, and adjunctions, and motivates the idea of representing objects by functors. In practice, it underlies approaches such as the functor of points in algebraic geometry and various developments in topos theory and higher category theory.

Historical note

The results are named after Nobuo Yoneda, who established them in 1954, and they remain foundational in category theory.