See homotopy on Wiktionary
{ "derived": [ { "_dis1": "0 0 0 0 0", "word": "cohomotopy" }, { "_dis1": "0 0 0 0 0", "word": "homotopic" }, { "_dis1": "0 0 0 0 0", "word": "homotopy category" }, { "_dis1": "0 0 0 0 0", "word": "homotopy class" }, { "_dis1": "0 0 0 0 0", "word": "homotopy equivalence" }, { "_dis1": "0 0 0 0 0", "word": "homotopy equivalent" }, { "_dis1": "0 0 0 0 0", "word": "homotopy fiber" }, { "_dis1": "0 0 0 0 0", "word": "homotopy fibre" }, { "_dis1": "0 0 0 0 0", "word": "homotopy group" }, { "_dis1": "0 0 0 0 0", "word": "homotopy invariant" }, { "_dis1": "0 0 0 0 0", "word": "homotopy relative to a subspace" }, { "_dis1": "0 0 0 0 0", "word": "homotopy theory" }, { "_dis1": "0 0 0 0 0", "word": "homotopy type" }, { "_dis1": "0 0 0 0 0", "word": "linear homotopy" }, { "_dis1": "0 0 0 0 0", "word": "regular homotopy" }, { "_dis1": "0 0 0 0 0", "word": "straight-line homotopy" } ], "etymology_templates": [ { "args": { "1": "en", "2": "grc", "3": "ὁμός", "4": "", "5": "same, similar" }, "expansion": "Ancient Greek ὁμός (homós, “same, similar”)", "name": "der" } ], "etymology_text": "From Ancient Greek ὁμός (homós, “same, similar”) + τόπος (tópos, “place”); earliest known use in print in 1922, Oswald Veblen, Analysis Situs.", "forms": [ { "form": "homotopies", "tags": [ "plural" ] } ], "head_templates": [ { "args": { "1": "~" }, "expansion": "homotopy (countable and uncountable, plural homotopies)", "name": "en-noun" } ], "lang": "English", "lang_code": "en", "pos": "noun", "related": [ { "_dis1": "0 0 0 0 0", "word": "homotopic" }, { "_dis1": "0 0 0 0 0", "word": "homeotopy group" }, { "_dis1": "0 0 0 0 0", "word": "homology" } ], "senses": [ { "categories": [ { "kind": "topical", "langcode": "en", "name": "Topology", "orig": "en:Topology", "parents": [ "Mathematics", "Formal sciences", "Sciences", "All topics", "Fundamental" ], "source": "w" } ], "examples": [ { "text": "The concept of homotopy represents a formalisation of the intuitive idea of a smooth deformation of one curve into another.", "type": "example" }, { "ref": "1998, Paul Sellick, “Space Exponents for Loop Spaces of Spheres”, in William G. Dwyer, editor, Stable and Unstable Homotopy, American Mathematical Society, page 279:", "text": "An integer M is called an exponent for the torsion of an abelian group G if M * (torsion of G) = 0. We say that M is a homotopy exponent for a space X if M is an exponent for πₖ (X) for all k.", "type": "quote" }, { "ref": "2001, F. R. Cohen, S. Gitler, “Loop-spaces of configuration spaces, braid-like groups, and knots”, in Jaume Aguadé, Carles Broto, Carles Casacuberta, editors, Cohomological Methods in Homotopy Theory, Springer (Birkhäuser), page 63:", "text": "A graded Lie algebra arises from these maps via the Samelson product in homotopy, the so-called homotopy Lie algebra which is discussed below.", "type": "quote" }, { "text": "2010, Vladimir G. Turaev, Homotopy Quantum Field Theory, European Mathematical Society, page xi,\nIn this monograph we apply the idea of a TQFT to maps from manifolds to topological spaces. This leads us to a notion of a (d + 1)-dimensional homotopy quantum field theory (HQFT) which may be described as a TQFT for closed oriented d-dimensional manifolds and compact oriented (d + 1)-dimensional cobordisms endowed with maps to a given space X." } ], "glosses": [ "A continuous deformation of one continuous function or map to another." ], "hyponyms": [ { "_dis1": "76 13 5 2 3", "sense": "continuous deformation", "word": "isotopy" }, { "_dis1": "76 13 5 2 3", "sense": "continuous deformation", "word": "regular homotopy" } ], "id": "en-homotopy-en-noun-mk5aiw6X", "links": [ [ "topology", "topology" ], [ "continuous", "continuous" ], [ "deformation", "deformation" ], [ "continuous function", "continuous function" ], [ "map", "map" ] ], "raw_glosses": [ "(topology) A continuous deformation of one continuous function or map to another." ], "tags": [ "countable", "uncountable" ], "topics": [ "mathematics", "sciences", "topology" ], "translations": [ { "_dis1": "56 17 4 11 12", "code": "da", "lang": "Danish", "sense": "(topology) continuous deformation of one continuous function to another", "tags": [ "common-gender" ], "word": "homotopi" }, { "_dis1": "56 17 4 11 12", "code": "fr", "lang": "French", "sense": "(topology) continuous deformation of one continuous function to another", "tags": [ "feminine" ], "word": "homotopie" }, { "_dis1": "56 17 4 11 12", "code": "el", "lang": "Greek", "roman": "omotopía", "sense": "(topology) continuous deformation of one continuous function to another", "tags": [ "feminine" ], "word": "ομοτοπία" }, { "_dis1": "56 17 4 11 12", "code": "hu", "lang": "Hungarian", "sense": "(topology) continuous deformation of one continuous function to another", "word": "homotópia" }, { "_dis1": "56 17 4 11 12", "code": "is", "lang": "Icelandic", "sense": "(topology) continuous deformation of one continuous function to another", "tags": [ "feminine" ], "word": "samtogun" }, { "_dis1": "56 17 4 11 12", "code": "ja", "lang": "Japanese", "sense": "(topology) continuous deformation of one continuous function to another", "word": "ホモトピー" }, { "_dis1": "56 17 4 11 12", "code": "ko", "lang": "Korean", "roman": "yeonsok byeonhyeong hamsu", "sense": "(topology) continuous deformation of one continuous function to another", "word": "연속 변형 함수" }, { "_dis1": "56 17 4 11 12", "code": "ko", "lang": "Korean", "roman": "homotopi", "sense": "(topology) continuous deformation of one continuous function to another", "word": "호모토피" }, { "_dis1": "56 17 4 11 12", "code": "fa", "lang": "Persian", "roman": "hemotopi", "sense": "(topology) continuous deformation of one continuous function to another", "word": "هموتوپی" }, { "_dis1": "56 17 4 11 12", "code": "pt", "lang": "Portuguese", "sense": "(topology) continuous deformation of one continuous function to another", "tags": [ "feminine" ], "word": "homotopia" }, { "_dis1": "56 17 4 11 12", "code": "es", "lang": "Spanish", "sense": "(topology) continuous deformation of one continuous function to another", "tags": [ "feminine" ], "word": "homotopía" }, { "_dis1": "56 17 4 11 12", "code": "uk", "lang": "Ukrainian", "roman": "homotopija", "sense": "(topology) continuous deformation of one continuous function to another", "tags": [ "feminine" ], "word": "гомотопія" }, { "_dis1": "56 17 4 11 12", "code": "vi", "lang": "Vietnamese", "sense": "(topology) continuous deformation of one continuous function to another", "word": "đồng luân" } ] }, { "categories": [], "glosses": [ "The relationship between two continuous functions where homotopy from one to the other is evident." ], "id": "en-homotopy-en-noun-YpP8NWnM", "raw_glosses": [ "(uncountable) The relationship between two continuous functions where homotopy from one to the other is evident." ], "tags": [ "uncountable" ] }, { "alt_of": [ { "extra": "the systematic study of homotopies and their equivalence classes", "word": "homotopy theory" } ], "categories": [ { "kind": "other", "name": "English links with redundant wikilinks", "parents": [ "Links with redundant wikilinks", "Entry maintenance" ], "source": "w" }, { "_dis": "4 29 48 14 6", "kind": "other", "name": "English entries with incorrect language header", "parents": [ "Entries with incorrect language header", "Entry maintenance" ], "source": "w+disamb" }, { "_dis": "3 18 66 9 4", "kind": "other", "name": "Entries with translation boxes", "parents": [], "source": "w+disamb" }, { "_dis": "4 29 46 15 6", "kind": "other", "name": "Pages with 1 entry", "parents": [], "source": "w+disamb" }, { "_dis": "2 32 50 13 3", "kind": "other", "name": "Pages with entries", "parents": [], "source": "w+disamb" }, { "_dis": "5 26 51 12 5", "kind": "other", "name": "Terms with Danish translations", "parents": [], "source": "w+disamb" }, { "_dis": "8 28 42 13 8", "kind": "other", "name": "Terms with French translations", "parents": [], "source": "w+disamb" }, { "_dis": "12 20 38 16 14", "kind": "other", "name": "Terms with Greek translations", "parents": [], "source": "w+disamb" }, { "_dis": "12 20 39 16 13", "kind": "other", "name": "Terms with Hungarian translations", "parents": [], "source": "w+disamb" }, { "_dis": "8 26 44 14 8", "kind": "other", "name": "Terms with Icelandic translations", "parents": [], "source": "w+disamb" }, { "_dis": "8 28 45 13 7", "kind": "other", "name": "Terms with Japanese translations", "parents": [], "source": "w+disamb" }, { "_dis": "8 26 44 14 8", "kind": "other", "name": "Terms with Korean translations", "parents": [], "source": "w+disamb" }, { "_dis": "8 26 44 14 8", "kind": "other", "name": "Terms with Persian translations", "parents": [], "source": "w+disamb" }, { "_dis": "7 29 42 13 8", "kind": "other", "name": "Terms with Portuguese translations", "parents": [], "source": "w+disamb" }, { "_dis": "6 29 47 13 6", "kind": "other", "name": "Terms with Spanish translations", "parents": [], "source": "w+disamb" }, { "_dis": "8 26 44 14 8", "kind": "other", "name": "Terms with Ukrainian translations", "parents": [], "source": "w+disamb" }, { "_dis": "8 26 44 14 8", "kind": "other", "name": "Terms with Vietnamese translations", "parents": [], "source": "w+disamb" } ], "glosses": [ "Ellipsis of homotopy theory. (the systematic study of homotopies and their equivalence classes)." ], "id": "en-homotopy-en-noun-s5Y-xV5B", "links": [ [ "homotopy theory", "homotopy theory#English" ], [ "equivalence class", "equivalence class#English" ] ], "raw_glosses": [ "(informal) Ellipsis of homotopy theory. (the systematic study of homotopies and their equivalence classes)." ], "tags": [ "abbreviation", "alt-of", "countable", "ellipsis", "informal", "uncountable" ] }, { "categories": [ { "kind": "topical", "langcode": "en", "name": "Topology", "orig": "en:Topology", "parents": [ "Mathematics", "Formal sciences", "Sciences", "All topics", "Fundamental" ], "source": "w" } ], "glosses": [ "A theory associating a system of groups with each topological space." ], "id": "en-homotopy-en-noun-RZXyIvoX", "links": [ [ "topology", "topology" ], [ "theory", "theory" ], [ "group", "group" ], [ "topological space", "topological space" ] ], "raw_glosses": [ "(topology) A theory associating a system of groups with each topological space." ], "tags": [ "countable", "uncountable" ], "topics": [ "mathematics", "sciences", "topology" ] }, { "categories": [ { "kind": "topical", "langcode": "en", "name": "Topology", "orig": "en:Topology", "parents": [ "Mathematics", "Formal sciences", "Sciences", "All topics", "Fundamental" ], "source": "w" } ], "glosses": [ "A system of groups associated with a topological space." ], "id": "en-homotopy-en-noun-EfBp7ZeX", "links": [ [ "topology", "topology" ] ], "raw_glosses": [ "(topology) A system of groups associated with a topological space." ], "tags": [ "countable", "uncountable" ], "topics": [ "mathematics", "sciences", "topology" ], "translations": [ { "_dis1": "5 1 4 38 52", "code": "fr", "lang": "French", "sense": "(topology) system of groups associated with a topological space", "tags": [ "feminine" ], "word": "homotopie" }, { "_dis1": "5 1 4 38 52", "code": "hu", "lang": "Hungarian", "sense": "(topology) system of groups associated with a topological space", "word": "homotópia" }, { "_dis1": "5 1 4 38 52", "code": "es", "lang": "Spanish", "sense": "(topology) system of groups associated with a topological space", "tags": [ "feminine" ], "word": "homotopía" } ] } ], "translations": [ { "_dis1": "5 3 8 47 38", "code": "fr", "lang": "French", "sense": "(topology) theory associating a system of groups with each topological space", "tags": [ "feminine" ], "word": "homotopie" }, { "_dis1": "5 3 8 47 38", "code": "hu", "lang": "Hungarian", "sense": "(topology) theory associating a system of groups with each topological space", "word": "homotópia" }, { "_dis1": "5 3 8 47 38", "code": "es", "lang": "Spanish", "sense": "(topology) theory associating a system of groups with each topological space", "tags": [ "feminine" ], "word": "homotopía" } ], "wikipedia": [ "Oswald Veblen", "homotopy" ], "word": "homotopy" }
{ "categories": [ "English countable nouns", "English entries with incorrect language header", "English lemmas", "English nouns", "English terms derived from Ancient Greek", "English uncountable nouns", "Entries with translation boxes", "Pages with 1 entry", "Pages with entries", "Terms with Danish translations", "Terms with French translations", "Terms with Greek translations", "Terms with Hungarian translations", "Terms with Icelandic translations", "Terms with Japanese translations", "Terms with Korean translations", "Terms with Persian translations", "Terms with Portuguese translations", "Terms with Spanish translations", "Terms with Ukrainian translations", "Terms with Vietnamese translations" ], "derived": [ { "word": "cohomotopy" }, { "word": "homotopic" }, { "word": "homotopy category" }, { "word": "homotopy class" }, { "word": "homotopy equivalence" }, { "word": "homotopy equivalent" }, { "word": "homotopy fiber" }, { "word": "homotopy fibre" }, { "word": "homotopy group" }, { "word": "homotopy invariant" }, { "word": "homotopy relative to a subspace" }, { "word": "homotopy theory" }, { "word": "homotopy type" }, { "word": "linear homotopy" }, { "word": "regular homotopy" }, { "word": "straight-line homotopy" } ], "etymology_templates": [ { "args": { "1": "en", "2": "grc", "3": "ὁμός", "4": "", "5": "same, similar" }, "expansion": "Ancient Greek ὁμός (homós, “same, similar”)", "name": "der" } ], "etymology_text": "From Ancient Greek ὁμός (homós, “same, similar”) + τόπος (tópos, “place”); earliest known use in print in 1922, Oswald Veblen, Analysis Situs.", "forms": [ { "form": "homotopies", "tags": [ "plural" ] } ], "head_templates": [ { "args": { "1": "~" }, "expansion": "homotopy (countable and uncountable, plural homotopies)", "name": "en-noun" } ], "hyponyms": [ { "sense": "continuous deformation", "word": "isotopy" }, { "sense": "continuous deformation", "word": "regular homotopy" } ], "lang": "English", "lang_code": "en", "pos": "noun", "related": [ { "word": "homotopic" }, { "word": "homeotopy group" }, { "word": "homology" } ], "senses": [ { "categories": [ "English terms with quotations", "English terms with usage examples", "en:Topology" ], "examples": [ { "text": "The concept of homotopy represents a formalisation of the intuitive idea of a smooth deformation of one curve into another.", "type": "example" }, { "ref": "1998, Paul Sellick, “Space Exponents for Loop Spaces of Spheres”, in William G. Dwyer, editor, Stable and Unstable Homotopy, American Mathematical Society, page 279:", "text": "An integer M is called an exponent for the torsion of an abelian group G if M * (torsion of G) = 0. We say that M is a homotopy exponent for a space X if M is an exponent for πₖ (X) for all k.", "type": "quote" }, { "ref": "2001, F. R. Cohen, S. Gitler, “Loop-spaces of configuration spaces, braid-like groups, and knots”, in Jaume Aguadé, Carles Broto, Carles Casacuberta, editors, Cohomological Methods in Homotopy Theory, Springer (Birkhäuser), page 63:", "text": "A graded Lie algebra arises from these maps via the Samelson product in homotopy, the so-called homotopy Lie algebra which is discussed below.", "type": "quote" }, { "text": "2010, Vladimir G. Turaev, Homotopy Quantum Field Theory, European Mathematical Society, page xi,\nIn this monograph we apply the idea of a TQFT to maps from manifolds to topological spaces. This leads us to a notion of a (d + 1)-dimensional homotopy quantum field theory (HQFT) which may be described as a TQFT for closed oriented d-dimensional manifolds and compact oriented (d + 1)-dimensional cobordisms endowed with maps to a given space X." } ], "glosses": [ "A continuous deformation of one continuous function or map to another." ], "links": [ [ "topology", "topology" ], [ "continuous", "continuous" ], [ "deformation", "deformation" ], [ "continuous function", "continuous function" ], [ "map", "map" ] ], "raw_glosses": [ "(topology) A continuous deformation of one continuous function or map to another." ], "tags": [ "countable", "uncountable" ], "topics": [ "mathematics", "sciences", "topology" ] }, { "categories": [ "English uncountable nouns" ], "glosses": [ "The relationship between two continuous functions where homotopy from one to the other is evident." ], "raw_glosses": [ "(uncountable) The relationship between two continuous functions where homotopy from one to the other is evident." ], "tags": [ "uncountable" ] }, { "alt_of": [ { "extra": "the systematic study of homotopies and their equivalence classes", "word": "homotopy theory" } ], "categories": [ "English ellipses", "English informal terms", "English links with redundant wikilinks" ], "glosses": [ "Ellipsis of homotopy theory. (the systematic study of homotopies and their equivalence classes)." ], "links": [ [ "homotopy theory", "homotopy theory#English" ], [ "equivalence class", "equivalence class#English" ] ], "raw_glosses": [ "(informal) Ellipsis of homotopy theory. (the systematic study of homotopies and their equivalence classes)." ], "tags": [ "abbreviation", "alt-of", "countable", "ellipsis", "informal", "uncountable" ] }, { "categories": [ "en:Topology" ], "glosses": [ "A theory associating a system of groups with each topological space." ], "links": [ [ "topology", "topology" ], [ "theory", "theory" ], [ "group", "group" ], [ "topological space", "topological space" ] ], "raw_glosses": [ "(topology) A theory associating a system of groups with each topological space." ], "tags": [ "countable", "uncountable" ], "topics": [ "mathematics", "sciences", "topology" ] }, { "categories": [ "en:Topology" ], "glosses": [ "A system of groups associated with a topological space." ], "links": [ [ "topology", "topology" ] ], "raw_glosses": [ "(topology) A system of groups associated with a topological space." ], "tags": [ "countable", "uncountable" ], "topics": [ "mathematics", "sciences", "topology" ] } ], "translations": [ { "code": "da", "lang": "Danish", "sense": "(topology) continuous deformation of one continuous function to another", "tags": [ "common-gender" ], "word": "homotopi" }, { "code": "fr", "lang": "French", "sense": "(topology) continuous deformation of one continuous function to another", "tags": [ "feminine" ], "word": "homotopie" }, { "code": "el", "lang": "Greek", "roman": "omotopía", "sense": "(topology) continuous deformation of one continuous function to another", "tags": [ "feminine" ], "word": "ομοτοπία" }, { "code": "hu", "lang": "Hungarian", "sense": "(topology) continuous deformation of one continuous function to another", "word": "homotópia" }, { "code": "is", "lang": "Icelandic", "sense": "(topology) continuous deformation of one continuous function to another", "tags": [ "feminine" ], "word": "samtogun" }, { "code": "ja", "lang": "Japanese", "sense": "(topology) continuous deformation of one continuous function to another", "word": "ホモトピー" }, { "code": "ko", "lang": "Korean", "roman": "yeonsok byeonhyeong hamsu", "sense": "(topology) continuous deformation of one continuous function to another", "word": "연속 변형 함수" }, { "code": "ko", "lang": "Korean", "roman": "homotopi", "sense": "(topology) continuous deformation of one continuous function to another", "word": "호모토피" }, { "code": "fa", "lang": "Persian", "roman": "hemotopi", "sense": "(topology) continuous deformation of one continuous function to another", "word": "هموتوپی" }, { "code": "pt", "lang": "Portuguese", "sense": "(topology) continuous deformation of one continuous function to another", "tags": [ "feminine" ], "word": "homotopia" }, { "code": "es", "lang": "Spanish", "sense": "(topology) continuous deformation of one continuous function to another", "tags": [ "feminine" ], "word": "homotopía" }, { "code": "uk", "lang": "Ukrainian", "roman": "homotopija", "sense": "(topology) continuous deformation of one continuous function to another", "tags": [ "feminine" ], "word": "гомотопія" }, { "code": "vi", "lang": "Vietnamese", "sense": "(topology) continuous deformation of one continuous function to another", "word": "đồng luân" }, { "code": "fr", "lang": "French", "sense": "(topology) theory associating a system of groups with each topological space", "tags": [ "feminine" ], "word": "homotopie" }, { "code": "hu", "lang": "Hungarian", "sense": "(topology) theory associating a system of groups with each topological space", "word": "homotópia" }, { "code": "es", "lang": "Spanish", "sense": "(topology) theory associating a system of groups with each topological space", "tags": [ "feminine" ], "word": "homotopía" }, { "code": "fr", "lang": "French", "sense": "(topology) system of groups associated with a topological space", "tags": [ "feminine" ], "word": "homotopie" }, { "code": "hu", "lang": "Hungarian", "sense": "(topology) system of groups associated with a topological space", "word": "homotópia" }, { "code": "es", "lang": "Spanish", "sense": "(topology) system of groups associated with a topological space", "tags": [ "feminine" ], "word": "homotopía" } ], "wikipedia": [ "Oswald Veblen", "homotopy" ], "word": "homotopy" }
Download raw JSONL data for homotopy meaning in All languages combined (8.7kB)
This page is a part of the kaikki.org machine-readable All languages combined dictionary. This dictionary is based on structured data extracted on 2024-12-15 from the enwiktionary dump dated 2024-12-04 using wiktextract (8a39820 and 4401a4c). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
If you use this data in academic research, please cite Tatu Ylonen: Wiktextract: Wiktionary as Machine-Readable Structured Data, Proceedings of the 13th Conference on Language Resources and Evaluation (LREC), pp. 1317-1325, Marseille, 20-25 June 2022. Linking to the relevant page(s) under https://kaikki.org would also be greatly appreciated.