The Yoneda lemma embeds a locally small category into a functor category
Image: Wassily Kandinsky, Public domain, via Wikimedia Commons
Yoneda lemma
The Yoneda lemma embeds a locally small category into a functor category
The Yoneda lemma allows for the embedding of any locally small category into a category of contravariant functors. This embedding is a powerful tool in category theory as it provides a way to study the original category through its functor representation. The lemma essentially states that an object in a category can be fully understood through its relationships with all other objects via morphisms.
Example
Consider a locally small category C with objects A, B, and C. The Yoneda lemma embeds C into a category of functors from C to the category of sets (Set). For instance, the functor Hom(A, -) maps any object X in C to the set Hom(A, X), capturing all morphisms from A to X.
Understanding the Yoneda lemma is crucial for modern developments in algebraic geometry and representation theory, as it provides a foundational tool for studying categories through their functor representations.
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