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Yoneda lemma (Theorem)

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.

Abelian Category Equivalence Lemma. 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 $hom_C (X,\^aˆ’) : \mathcal{C} \to {\bf Sets}$. One has also that the assignment $X \mapsto hom_C (X,\^aˆ’)$ 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.

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



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See Also: fully faithful functor, Abelian category, B-mod category equivalence theorem, Morita equivalence

Also defines:  Yoneda functor, hom-functor, Abelian category equivalence lemma
Keywords:  categorical physics, Yoneda lemma

Cross-references: isomorphisms, proposition, object, functor category, category, functor, surjective, fully faithful functors, Abelian categories, category theory
There are 2 references to this object.

This is version 13 of Yoneda lemma, born on 2009-06-15, modified 2009-06-15.
Object id is 797, canonical name is YonedaLemma.
Accessed 990 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
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