English Dictionary

Yoneda lemma

名詞

(category theory) 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.)

日本語の意味

圏論におけるヨネダの補題（Yoneda lemma）とは、ある圏𝒞とその中の対象Aが与えられた場合、Aによって表現されるホム関手Hと、必ずしも表現可能でない任意の関手F（集合から圏𝒞への関手）との間で、自然変換の集合Nat(H, F)とF(A)の間に自然同型が存在するという主張である。

By the Yoneda lemma, any natural transformation from the Hom-functor represented by A to a functor F is uniquely determined by the element α_A(id_A) in F(A).

ヨネダの補題により、A によって表される Hom 関手から関手 F への任意の自然変換は、F(A) にある元 α_A(id_A) によって一意に決定されます。

意味（1）

(category theory) 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.)

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 :-)