PlanetPhysics/Yoneda Lemma

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Yoneda lemma

Let us introduce first a basic lemma in [[../TrivialGroupoid/|category theory]] that links the equivalence of two [[../AbelianCategory2/|abelian categories]] to certain [[../FullyFaithfulFunctor2/|fully faithful functors]].

{\mathbf Abelian Category Equivalence Lemma.} Let Failed to parse (syntax error): {\displaystyle \mathcal{A'' } and $ℬ$ be any two Abelian categories, and also let $F:𝒜\to ℬ$ be an exact, fully faithful, essentially [[../BCConjecture/|surjective]] [[../TrivialGroupoid/|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 [[../Cod/|category]] of sets. The functors $F:𝒞\to 𝐒ets$, for any category $𝒞$, form a [[../TrivialGroupoid/|functor category]] $𝐅unct(𝒞,𝐒ets)$ (also written as $[𝒞,𝐒ets]$. Then, any [[../TrivialGroupoid/|object]] $X\in 𝒞$ 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 $F_y:𝒞\to 𝐅unct(𝒞,𝐒ets)$.

One of the most commonly used results in category theory for establishing an equivalence of categories is provided by the following [[../Predicate/|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 [[../FullyFaithfulFunctor2/|fully faithful functor]] because it induces [[../IsomorphicObjectsUnderAnIsomorphism/|isomorphisms]] on the Hom sets.}