Meaning of Yoneda lemma | Babel Free

Definitions

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.)

Equivalents

Examples

“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.”

“• 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.”

“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 :-)”

CEFR level