Yoneda lemma

English

Etymology

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

Noun

Yoneda lemma

1. (category theory) Given a category ${\mathcal {C}}$ with an object A, let H be a hom functor represented by A, and let F be any functor (not necessarily representable) from ${\mathcal {C}}$ 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 $\alpha$ from H to F is determined by what $\alpha _{A}({\mbox{id}}_{A})$ is.)

   As a corollary of the Yoneda lemma, given a pair of contravariant hom functors ${\mbox{Hom}}(-,A)$ and ${\mbox{Hom}}(-,B)$, then any natural transformation $\alpha$ from ${\mbox{Hom}}(-,A)$ to ${\mbox{Hom}}(-,B)$ is determined by the choice of some function $f:A\rightarrow B$ to map the identity ${\mbox{id}}_{A}:A\rightarrow A$ to, by the component $\alpha _{A}:{\mbox{Hom}}(A,A)\rightarrow {\mbox{Hom}}(A,B)$ of $\alpha$. This implies that the Yoneda functor is fully faithful, which in turn implies that Yoneda embeddings are possible.

   • Yoneda Lemma: Nat(Hom(A,–), F) ≅ F(A)
   ∴ Nat(Hom(A,–), Hom(B,–)) ≅ Hom(B,A)
   ∴ A ≅ B iff Hom(A,–) ≅ 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.

   • 2020, Emily Riehl, quoting Fred E. J. Linton, The Yoneda lemma in the category of Matrices‎^[2]:

     And then there's the Yoneda Lemma embodied in the classical Gaussian row reduction operation, that a given row reduction operation (on matrices with say k rows) being a "natural" operation (in the sense of natural transformations) is just multiplication (on the appropriate side) by the effect of that operation on the k-by-k identity matrix.
     
     And dually for column-reduction operations :-)

Translations

theorem which states that there is a natural isomorphism...

• Chinese:

  Mandarin: 米田引理 (Mǐtián yǐnlǐ)

• French: lemme de Yoneda m
• German: Lemma von Yoneda n
• Italian: lemma di Yoneda m
• Japanese: 米田の補題 (Yoneda no hodai)
• Korean: 요네다 보조정리 (Yoneda bojojeongni)
• Portuguese: lema de Yoneda m
• Spanish: lema de Yoneda m