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

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

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