Yoneda lemma

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 $ \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 $ {\bf Sets}$ be the category of sets. The functors $ F: \mathcal{C} \to {\bf Sets}$ , for any category $ \mathcal{C}$ , form a functor category $ {\bf Funct}(\mathcal{C},{\bf Sets})$ (also written as $ [\mathcal{C},{\bf Sets}]$ . Then, any object $ X \in \mathcal{C}$ gives rise to the functor $ hom_C (X,−) : \mathcal{C} \to {\bf Sets}$ . One has also that the assignment $ X \mapsto hom_C (X,−)$ extends to a natural contravariant functor $ F_y: \mathcal{C} \to {\bf Funct}(\mathcal{C},{\bf 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 $ F_y: \mathcal{C} \to {\bf Funct}(\mathcal{C},{\bf Sets})$ is a fully faithful functor because it induces isomorphisms on the Hom sets.



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