Yoneda lemma

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Yoneda lemma

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

Yoneda lemma

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

   As a corollary of the Yoneda lemma, given a pair of contravariant hom functors ${\displaystyle {\mbox{Hom}}(-,A)}$ and ${\displaystyle {\mbox{Hom}}(-,B)}$, then any natural transformation ${\displaystyle \alpha }$ from ${\displaystyle {\mbox{Hom}}(-,A)}$ to ${\displaystyle {\mbox{Hom}}(-,B)}$ is determined by the choice of some function ${\displaystyle f:A\rightarrow B}$ to map the identity ${\displaystyle {\mbox{id}}_{A}:A\rightarrow A}$ to, by the component ${\displaystyle \alpha _{A}:{\mbox{Hom}}(A,A)\rightarrow {\mbox{Hom}}(A,B)}$ of ${\displaystyle \alpha }$. 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.

   ^[1]

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

     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 :-)