yoneda lemma
Word Rarity
Very rare
Frequency: Zipf frequency 1.39 in modern English
19
Scrabble Score
11
Letters
2
Syllables
8
Unique Chars
Meaning
noun
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.)
Example Sentences
- "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."
- "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."
- "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 :-)"