See Yoneda lemma in All languages combined, or Wiktionary
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{
"etymology_text": "Lemma named after the Japanese mathematician Nobuo Yoneda (1930–1996).",
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"text": "As a corollary of the Yoneda lemma, given a pair of contravariant hom functors #x5C;mbox#x7B;Hom#x7D;(-,A) and #x5C;mbox#x7B;Hom#x7D;(-,B), then any natural transformation #x5C;alpha from #x5C;mbox#x7B;Hom#x7D;(-,A) to #x5C;mbox#x7B;Hom#x7D;(-,B) is determined by the choice of some function f#x3A;A#x5C;rightarrowB to map the identity #x5C;mbox#x7B;id#x7D;#x5F;A#x3A;A#x5C;rightarrowA to, by the component #x5C;alpha#x5F;A#x3A;#x5C;mbox#x7B;Hom#x7D;(A,A)#x5C;rightarrow#x5C;mbox#x7B;Hom#x7D;(A,B) of #x5C;alpha. This implies that the Yoneda functor is fully faithful, which in turn implies that Yoneda embeddings are possible.",
"type": "example"
},
{
"ref": "• Yoneda Lemma: Nat(Hom(A,–), F) ≅ F(A)",
"text": "∴ Nat(Hom(A,–), Hom(B,–)) ≅ Hom(B,A)\n∴ A ≅ B iff Hom(A,–) ≅ Hom(B,–)\ni.e. A is isomorphic to B if and only if A's network of relations is isomorphic to B's network of relations.",
"type": "example"
},
{
"ref": "2020, Emily Riehl, quoting Fred E. J. Linton, The Yoneda lemma in the category of Matrices",
"text": "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.\nAnd dually for column-reduction operations :-)",
"type": "quotation"
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"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.)"
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"(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.)"
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"translations": [
{
"code": "cmn",
"lang": "Chinese Mandarin",
"roman": "Mǐtián yǐnlǐ",
"sense": "theorem which states that there is a natural isomorphism...",
"word": "米田引理"
},
{
"code": "fr",
"lang": "French",
"sense": "theorem which states that there is a natural isomorphism...",
"tags": [
"masculine"
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"word": "lemme de Yoneda"
},
{
"code": "de",
"lang": "German",
"sense": "theorem which states that there is a natural isomorphism...",
"tags": [
"neuter"
],
"word": "Lemma von Yoneda"
},
{
"code": "it",
"lang": "Italian",
"sense": "theorem which states that there is a natural isomorphism...",
"tags": [
"masculine"
],
"word": "lemma di Yoneda"
},
{
"code": "ja",
"english": "Yoneda no hodai",
"lang": "Japanese",
"sense": "theorem which states that there is a natural isomorphism...",
"word": "米田の補題"
},
{
"code": "ko",
"english": "Yoneda bojojeongni",
"lang": "Korean",
"sense": "theorem which states that there is a natural isomorphism...",
"word": "요네다 보조정리"
},
{
"code": "pt",
"lang": "Portuguese",
"sense": "theorem which states that there is a natural isomorphism...",
"tags": [
"masculine"
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"word": "lema de Yoneda"
},
{
"code": "es",
"lang": "Spanish",
"sense": "theorem which states that there is a natural isomorphism...",
"tags": [
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"word": "lema de Yoneda"
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"word": "Yoneda lemma"
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"text": "As a corollary of the Yoneda lemma, given a pair of contravariant hom functors #x5C;mbox#x7B;Hom#x7D;(-,A) and #x5C;mbox#x7B;Hom#x7D;(-,B), then any natural transformation #x5C;alpha from #x5C;mbox#x7B;Hom#x7D;(-,A) to #x5C;mbox#x7B;Hom#x7D;(-,B) is determined by the choice of some function f#x3A;A#x5C;rightarrowB to map the identity #x5C;mbox#x7B;id#x7D;#x5F;A#x3A;A#x5C;rightarrowA to, by the component #x5C;alpha#x5F;A#x3A;#x5C;mbox#x7B;Hom#x7D;(A,A)#x5C;rightarrow#x5C;mbox#x7B;Hom#x7D;(A,B) of #x5C;alpha. This implies that the Yoneda functor is fully faithful, which in turn implies that Yoneda embeddings are possible.",
"type": "example"
},
{
"ref": "• Yoneda Lemma: Nat(Hom(A,–), F) ≅ F(A)",
"text": "∴ Nat(Hom(A,–), Hom(B,–)) ≅ Hom(B,A)\n∴ A ≅ B iff Hom(A,–) ≅ Hom(B,–)\ni.e. A is isomorphic to B if and only if A's network of relations is isomorphic to B's network of relations.",
"type": "example"
},
{
"ref": "2020, Emily Riehl, quoting Fred E. J. Linton, The Yoneda lemma in the category of Matrices",
"text": "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.\nAnd dually for column-reduction operations :-)",
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"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.)"
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{
"code": "cmn",
"lang": "Chinese Mandarin",
"roman": "Mǐtián yǐnlǐ",
"sense": "theorem which states that there is a natural isomorphism...",
"word": "米田引理"
},
{
"code": "fr",
"lang": "French",
"sense": "theorem which states that there is a natural isomorphism...",
"tags": [
"masculine"
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"word": "lemme de Yoneda"
},
{
"code": "de",
"lang": "German",
"sense": "theorem which states that there is a natural isomorphism...",
"tags": [
"neuter"
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"word": "Lemma von Yoneda"
},
{
"code": "it",
"lang": "Italian",
"sense": "theorem which states that there is a natural isomorphism...",
"tags": [
"masculine"
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"word": "lemma di Yoneda"
},
{
"code": "ja",
"english": "Yoneda no hodai",
"lang": "Japanese",
"sense": "theorem which states that there is a natural isomorphism...",
"word": "米田の補題"
},
{
"code": "ko",
"english": "Yoneda bojojeongni",
"lang": "Korean",
"sense": "theorem which states that there is a natural isomorphism...",
"word": "요네다 보조정리"
},
{
"code": "pt",
"lang": "Portuguese",
"sense": "theorem which states that there is a natural isomorphism...",
"tags": [
"masculine"
],
"word": "lema de Yoneda"
},
{
"code": "es",
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"sense": "theorem which states that there is a natural isomorphism...",
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"word": "lema de Yoneda"
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"word": "Yoneda lemma"
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