Yoneda lemma

名词 n.
中文释义

英文释义

名词 n.
  1. Given a category 𝒞 with an object A, let H be a hom functor represented by A, and let F be any functor (not necessarily representable) from 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 α from H to F is determined by what α_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 f#58;A#92;rightarrowB to map the identity #92;mbox#123;id#125;#95;A#58;A#92;rightarrowA to, by the component #92;alpha#95;A#58;#92;mbox#123;Hom#125;(A,A)#92;rightarrow#92;mbox#123;Hom#125;(A,B) of #92;alpha. This implies that the Yoneda functor is fully faithful, which in turn implies that Yoneda embeddings are possible.

词源

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