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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \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 ${\mathbf {S} ets}$ be the category of sets. The functors $F:{\mathcal {C}}\to {\mathbf {S} ets}$, for any category ${\mathcal {C}}$, form a functor category ${\mathbf {F} unct}({\mathcal {C}},{\mathbf {S} ets})$ (also written as $[{\mathcal {C}},{\mathbf {S} ets}]$. Then, any object $X\in {\mathcal {C}}$ 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 $F_{y}:{\mathcal {C}}\to {\mathbf {F} unct}({\mathcal {C}},{\mathbf {S} ets})$.

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 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.}