The Yoneda Lemma

Quoting Wikipedia on the Yoneda lemma (chainging A to a and C to A so that we can use unicode superscripts and subscripts): In mathematics, specifically in category theory, the Yoneda lemma is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a particular kind of category with just one object). It allows the embedding of any category into a category of functors (contravariant set-valued functors) defined on that category. It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category. It is an important tool that underlies several modern developments in algebraic geometry and representation theory. It is named after Nobuo Yoneda.

Generalities

The Yoneda lemma suggests that instead of studying the (locally small) category A, one should study the category of all functors of A into Set (the category of sets with functions as morphisms). Set is a category we understand well, and a functor of A into Set can be seen as a "representation" of A in terms of known structures. The original category A is contained in this functor category, but new objects appear in the functor category which were absent and "hidden" in A. Treating these new objects just like the old ones often unifies and simplifies the theory. This approach is akin to (and in fact generalizes) the common method of studying a ring by investigating the modules over that ring. The ring takes the place of the category A, and the category of modules over the ring is a category of functors defined on A.

Formal statement

General version

Yoneda's lemma concerns functors from a fixed category A to the category of sets, Set. If A is a locally small category (i.e. the hom-sets are actual sets and not proper classes), then each object a of A gives rise to a natural functor to Set called a hom-functor. This functor is denoted: hᵃ = Hom(a, ─). The (covariant) hom-functor hᵃ sends x to the set of morphisms Hom(a, x) and sends a morphism f from x to y to the morphism f ∘ ─ (composition with f on the left) that sends a morphism g in Hom(a, x) to the morphism f ∘ g in Hom(a, y). That is, f ↦ Hom(a, f) = ⟦ Hom(a, x) ∋ g ↦ f ∘ g ∈ Hom(a,y) ⟧.

The (co)yoneda functors A → (Aᵒᵖ → set)

The (co)yoneda lemma

Let F be an arbitrary functor from A to Set. Then Yoneda's lemma says that:

For each object a of A,

the natural transformations from hᵃ to F are in one-to-one correspondence with the elements of F(a). That is, Nat(hᵃ, F) ≅ F(a). Moreover this isomorphism is natural in a and F when both sides are regarded as functors from Setᴬ × A to Set. Given a natural transformation Φ from hᵃ to F, the corresponding element of F(a) is u = Φₐ(idₐ).

There is a contravariant version of Yoneda's lemma which concerns contravariant functors from A to Set. This version involves the contravariant hom-functor hₐ = Hom(─, A), which sends x to the hom-set Hom(x, a). Given an arbitrary contravariant functor G from A to Set, Yoneda's lemma asserts that Nat(hₐ, G) ≅ G(a).

Let F be an arbitrary functor from A to Set. Then Yoneda's lemma says that:

For each object a of A,

the natural transformations from hᵃ to F are in one-to-one correspondence with the elements of F(a). That is, Nat(hᵃ, F) ≅ F(a). Moreover this isomorphism is natural in a and F when both sides are regarded as functors from Setᴬ × A to Set. Given a natural transformation Φ from hᵃ to F, the corresponding element of F(a) is u = Φₐ(idₐ).

The Yoneda embedding

An important special case of Yoneda's lemma is when the functor F from A to Set is another hom-functor hᵇ. In this case, the covariant version of Yoneda's lemma states that Nat(hᵃ, hᵇ) ≅ Hom(b, a). That is, natural transformations between hom-functors are in one-to-one correspondence with morphisms (in the reverse direction) between the associated objects. Given a morphism f : b → a the associated natural transformation is denoted Hom(f, ─). Mapping each object a in A to its associated hom-functor hᵃ= Hom(a, ─) and each morphism f : B → A to the corresponding natural transformation Hom(f, ─) determines a contravariant functor h⁻ from A to Setᴬ, the functor category of all (covariant) functors from A to Set. One can interpret h⁻ as a covariant functor: h⁻ : Aᵒᵖ → Setᴬ. The meaning of Yoneda's lemma in this setting is that the functor h⁻ is fully faithful, and therefore gives an embedding of Aᵒᵖ in the category of functors to Set. The collection of all functors {hᵃ, a in A} is a subcategory of Set̂ᴬ. Therefore, Yoneda embedding implies that the category Aᵒᵖ is isomorphic to the category {hᵃ, a in A}. The contravariant version of Yoneda's lemma states that Nat(hₐ, h_b) ≅ Hom(a, b). Therefore, h₋ gives rise to a covariant functor from A to the category of contravariant functors to Set: h₋ : A → Set⁽⁽ᴬ⁾ᵒᵖ⁾. Yoneda's lemma then states that any locally small category A can be embedded in the category of contravariant functors from A to Set via h₋. This is called the Yoneda embedding.