Yoneda lemma
YonedaLemma
2009-06-15 06:50:56
2009-06-15 08:35:52
Theorem
Yoneda functor
hom-functor
categorical physics
Yoneda lemma
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\section{Yoneda lemma}
Let us introduce first a lemma that links the equivalence of two
Abelian categories to certain fully faithful functors.
{\bf Abelian Category Equivalence Lemma.}
{\em 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.
{\bf Yoneda Lemma.}{\em The functor $F_y: \mathcal{C} \to {\bf Funct}(\mathcal{C},{\bf Sets})$ induces isomorphisms on the Hom sets, and therefore, it is a fully faithful functor.}