1 Yoneda lemma
Let us introduce first a basic lemma in category theory that links the equivalence of two abelian
categories to certain fully faithful functors.
Abelian Category Equivalence Lemma. Let 𝒜 and ℬ be any two Abelian categories, and
also let F : 𝒜→ℬ be an exact, fully faithful, essentially surjective functor. faithful,
essentially surjective functor. Then F is an equivalence of Abelian categories 𝒜 and
ℬ.
The next step is to define the hom-functors. Let Sets be the category of sets. The functors
F : 𝒞→ Sets, for any category 𝒞, form a functor category Funct(𝒞,Sets) (also written as [𝒞,Sets].
Then, any object X ∈𝒞 gives rise to the functor homC(X,â ˆ ′) : 𝒞→ Sets. One has
also that the assignment X
homC(X,â ˆ ′) extends to a natural contravariant functor
Fy : 𝒞→ Funct(𝒞,Sets).
One of the most commonly used results in category theory for establishing an equivalence of
categories is provided by the following proposition.
Yoneda Lemma.The functor Fy : 𝒞→ Funct(𝒞,Sets) is a fully faithful functor because it induces
isomorphisms on the Hom sets.
