PlanetPhysics/Yoneda Lemma

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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.

{\mathbf Abelian Category Equivalence Lemma.} Let $$\mathcal{A'' $$ and $$\mathcal{B}$$ be any two Abelian categories, and also let $$F: \mathcal{A} \to \mathcal{B}$$ be an exact, fully faithful, essentially surjective functor. faithful, essentially surjective functor. Then $$F$$ is an equivalence of Abelian categories $$\mathcal{A}$$ and $$\mathcal{B}$$}.

The next step is to define the hom-functors. Let $${\mathbf Sets}$$ be the category of sets. The functors $$F: \mathcal{C} \to {\mathbf Sets}$$, for any category $$\mathcal{C}$$, form a functor category $${\mathbf Funct}(\mathcal{C},{\mathbf Sets})$$ (also written as $$[\mathcal{C},{\mathbf Sets}]$$. Then, any object $$X \in \mathcal{C}$$ gives rise to the functor $$hom_C (X,âˆ’) : \mathcal{C} \to {\mathbf Sets}$$. One has also that the assignment $$X \mapsto hom_C (X,âˆ’)$$ extends to a natural contravariant functor $$F_y: \mathcal{C} \to {\mathbf Funct}(\mathcal{C},{\mathbf Sets})$$.

One of the most commonly used results in category theory for establishing an equivalence of categories is provided by the following proposition.

{\mathbf Yoneda Lemma.} The functor $$F_y: \mathcal{C'' \to {\mathbf Funct}(\mathcal{C},{\mathbf Sets})$$ is a fully faithful functor because it induces isomorphisms on the Hom sets.}