"homotopy" meaning in All languages combined

See homotopy on Wiktionary

Noun [English]

Forms: homotopies [plural]
Etymology: From Ancient Greek ὁμός (homós, “same, similar”) + τόπος (tópos, “place”); earliest known use in print in 1922, Oswald Veblen, Analysis Situs. Etymology templates: {{der|en|grc|ὁμός||same, similar}} Ancient Greek ὁμός (homós, “same, similar”) Head templates: {{en-noun|~}} homotopy (countable and uncountable, plural homotopies)
  1. (topology) A continuous deformation of one continuous function or map to another. Tags: countable, uncountable Categories (topical): Topology Hyponyms (continuous deformation): isotopy, regular homotopy Translations ((topology) continuous deformation of one continuous function to another): homotopi [common-gender] (Danish), homotopie [feminine] (French), ομοτοπία (omotopía) [feminine] (Greek), homotópia (Hungarian), samtogun [feminine] (Icelandic), ホモトピー (Japanese), 연속 변형 함수 (yeonsok byeonhyeong hamsu) (Korean), 호모토피 (homotopi) (Korean), هموتوپی (hemotopi) (Persian), homotopia [feminine] (Portuguese), homotopía [feminine] (Spanish), гомотопія (homotopija) [feminine] (Ukrainian), đồng luân (Vietnamese)
    Sense id: en-homotopy-en-noun-mk5aiw6X Topics: mathematics, sciences, topology Disambiguation of 'continuous deformation': 76 13 5 2 3 Disambiguation of '(topology) continuous deformation of one continuous function to another': 56 17 4 11 12
  2. (uncountable) The relationship between two continuous functions where homotopy from one to the other is evident. Tags: uncountable
    Sense id: en-homotopy-en-noun-YpP8NWnM
  3. (informal) Ellipsis of homotopy theory. (the systematic study of homotopies and their equivalence classes). Tags: abbreviation, alt-of, countable, ellipsis, informal, uncountable Alternative form of: homotopy theory (extra: the systematic study of homotopies and their equivalence classes)
    Sense id: en-homotopy-en-noun-s5Y-xV5B Categories (other): English links with redundant wikilinks, English entries with incorrect language header, 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 Disambiguation of English entries with incorrect language header: 4 29 48 14 6 Disambiguation of Entries with translation boxes: 3 18 66 9 4 Disambiguation of Pages with 1 entry: 4 29 46 15 6 Disambiguation of Pages with entries: 2 32 50 13 3 Disambiguation of Terms with Danish translations: 5 26 51 12 5 Disambiguation of Terms with French translations: 8 28 42 13 8 Disambiguation of Terms with Greek translations: 12 20 38 16 14 Disambiguation of Terms with Hungarian translations: 12 20 39 16 13 Disambiguation of Terms with Icelandic translations: 8 26 44 14 8 Disambiguation of Terms with Japanese translations: 8 28 45 13 7 Disambiguation of Terms with Korean translations: 8 26 44 14 8 Disambiguation of Terms with Persian translations: 8 26 44 14 8 Disambiguation of Terms with Portuguese translations: 7 29 42 13 8 Disambiguation of Terms with Spanish translations: 6 29 47 13 6 Disambiguation of Terms with Ukrainian translations: 8 26 44 14 8 Disambiguation of Terms with Vietnamese translations: 8 26 44 14 8
  4. (topology) A theory associating a system of groups with each topological space. Tags: countable, uncountable Categories (topical): Topology
    Sense id: en-homotopy-en-noun-RZXyIvoX Topics: mathematics, sciences, topology
  5. (topology) A system of groups associated with a topological space. Tags: countable, uncountable Categories (topical): Topology Translations ((topology) system of groups associated with a topological space): homotopie [feminine] (French), homotópia (Hungarian), homotopía [feminine] (Spanish)
    Sense id: en-homotopy-en-noun-EfBp7ZeX Topics: mathematics, sciences, topology Disambiguation of '(topology) system of groups associated with a topological space': 5 1 4 38 52
The following are not (yet) sense-disambiguated
Related terms: homotopic, homeotopy group, homology Translations ((topology) theory associating a system of groups with each topological space): homotopie [feminine] (French), homotópia (Hungarian), homotopía [feminine] (Spanish)
Derived forms: cohomotopy, homotopic, homotopy category, homotopy class, homotopy equivalence, homotopy equivalent, homotopy fiber, homotopy fibre, homotopy group, homotopy invariant, homotopy relative to a subspace, homotopy theory, homotopy type, linear homotopy, regular homotopy, straight-line homotopy Disambiguation of '(topology) theory associating a system of groups with each topological space': 5 3 8 47 38

Inflected forms

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      "word": "cohomotopy"
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      "word": "homotopy category"
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      "word": "linear homotopy"
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    {
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      "word": "regular homotopy"
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    {
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          "text": "The concept of homotopy represents a formalisation of the intuitive idea of a smooth deformation of one curve into another.",
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          "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.",
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          "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.",
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          "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."
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      "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"
}

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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.

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