PlanetMath: Yoneda lemma

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Yoneda lemma

(Theorem)

If $\mathcal{C}$ is a category, write $\hat{\mathcal{C}}$ for the category of contravariant functors from $\mathcal{C}$ to ${\bf Sets}$ , the category of sets. The morphisms in $\hat{\mathcal{C}}$ are natural transformations of functors.

(To avoid set theoretical concerns, one can take a universe $\mathcal{U}$ and take all categories to be $\mathcal{U}$ -small.)

For any object of $\mathcal{C}$ , $h_X = {\rm Hom}(-,X)$ is a contravariant functor from $\mathcal{C}$ to ${\bf Sets}$ , and therefore is an object of $\hat{\mathcal{C}}$ .

Yoneda Lemma says that $X\mapsto h_X$ is a covariant functor $\mathcal{C}\to\hat{\mathcal{C}}$ , which embeds $\mathcal{C}$ faithfully as a full subcategory of $\hat{\mathcal{C}}$ . This embedding is called the Yoneda embedding.

"Yoneda lemma" is owned by rspuzio. [ full author list (4) | owner history (3) ]

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Other names:

Yoneda's Lemma

Also defines:

Yoneda embedding

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Cross-references: embedding, full subcategory, faithfully, object, universe, natural transformations, morphisms, category of sets, contravariant functors, category
There are 4 references to this entry.

This is version 9 of Yoneda lemma, born on 2002-02-02, modified 2008-05-01.
Object id is 1638, canonical name is YonedaEmbedding.
Accessed 5585 times total.

Classification:

AMS MSC:

18A25 (Category theory; homological algebra :: General theory of categories and functors :: Functor categories, comma categories)

Pending Errata and Addenda

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