PlanetPhysics/Yoneda Lemma

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Yoneda lemma

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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 Failed to parse (syntax error): {\displaystyle \mathcal{A'' } and   be any two Abelian categories, and also let   be an exact, fully faithful, essentially surjective functor. faithful, essentially surjective functor. Then   is an equivalence of Abelian categories   and  }.

The next step is to define the hom-functors. Let   be the category of sets. The functors  , for any category  , form a functor category   (also written as  . Then, any object   gives rise to the functor Failed to parse (syntax error): {\displaystyle hom_C (X,−) : \mathcal{C} \to {\mathbf Sets}} . One has also that the assignment Failed to parse (syntax error): {\displaystyle X \mapsto hom_C (X,−)} extends to a natural contravariant functor  .

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 Failed to parse (syntax error): {\displaystyle F_y: \mathcal{C'' \to {\mathbf Funct}(\mathcal{C},{\mathbf Sets})} is a fully faithful functor because it induces isomorphisms on the Hom sets.}