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Yoneda lemma

Yoneda lemma noun: given a category π’ž with an object A, let H be a hom functor represented by A, and let F be any functor (not necessarily representable) from π’ž to Sets, then there is a natural isomorphism between Nat(H,F), the set of natural transformations from H to F, and the set F(A). (Any natural transformation Ξ± from H to F is determined by what Ξ±_A( mbox id_A) is.).

Noun

Yoneda lemma

  1. Given a category π’ž with an object A, let H be a hom functor represented by A, and let F be any functor (not necessarily representable) from π’ž to Sets, then there is a natural isomorphism between Nat(H,F), the set of natural transformations from H to F, and the set F(A). (Any natural transformation Ξ± from H to F is determined by what Ξ±_A( mbox id_A) is.)
    • As a corollary of the Yoneda lemma, given a pair of contravariant hom functors #92;mbox#123;Hom#125;(-,A) and #92;mbox#123;Hom#125;(-,B), then any natural transformation #92;alpha from #92;mbox#123;Hom#125;(-,A) to #92;mbox#123;Hom#125;(-,B) is determined by the choice of some function...
    • β€’ Yoneda Lemma: Nat(Hom(A,–), F) β‰… F(A)
      ∴ Nat(Hom(A,–), Hom(B,–)) β‰… Hom(B,A)
      ∴ A β‰… B iff Hom(A,–) β‰… Hom(B,–)
      i.e. A is isomorphic to B if and only if A's network of relations is isomorphic to B's network of relations.

Origin of Yoneda lemma

Lemma named after the Japanese mathematician Nobuo Yoneda (1930–1996).

Questions about Yoneda lemma

What does Yoneda lemma mean?
Yoneda lemma means given a category π’ž with an object A, let H be a hom functor represented by A, and let F be any functor (not necessarily representable) from π’ž to Sets, then there is a natural isomorphism between Nat(H,F), the set of natural transformations from H to F, and the set F(A). (Any natural transformation Ξ± from H to F is determined by what Ξ±_A( mbox id_A) is.).
What part of speech is Yoneda lemma?
Yoneda lemma is a noun.

Words near Yoneda lemma in the dictionary

  1. Yoneda
  2. Yoneda embedding
  3. Yoneda embeddings
  4. Yoneda functor
  5. Yoneda functors
  6. Yoneda lemma
  7. Yonezawa
  8. yong
  9. yong bell
  10. yong bells
  11. yong tao foo

Source and license

Adapted from Wiktionary: Yoneda lemma (page history and authors). Text available under the Creative Commons Attribution-ShareAlike 4.0 license. Word.ws reformatted the entry and added links. Wiktionary data of 2026-09-02.

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