English Dictionary

Yoneda lemma

noun

(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.)

The Yoneda lemma is a fundamental result in category theory.

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 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 :-)