Yoneda lemma
nounEtymology
Lemma named after the Japanese mathematician Nobuo Yoneda (1930β1996).
Definitions
Given a category π with an object A, let H be a hom functor represented by A, and let Fβ¦
Given a category π with an object A, let H be a hom functor represented by A, and let F be any functor (not necessarily representable) from π 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 Ξ± from H to F is determined by what Ξ±_A( mbox id_A) is.)
- β’ Yoneda Lemma: Nat(Hom(A,β), F) β F(A) β΄ Nat(Hom(A,β), Hom(B,β)) β Hom(B,A) β΄ A β B iff Hom(A,β) β Hom(B,β) i.e. A is isomorphic to B if and only if A's network of relations is isomorphic to B's network of relations.
The neighborhood
Vish β recursive loop
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sense glosses and etymology drawn from English Wiktionary Β· source Β· CC-BY-SA