"Kan extension" meaning in All languages combined

{
  "etymology_text": "Named after Jewish and Dutch mathematician Daniel M. Kan (1927–2013), who constructed certain (Kan) extensions using limits in 1960.",
  "forms": [
    {
      "form": "Kan extensions",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "Kan extension (plural Kan extensions)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Entries with translation boxes",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with 1 entry",
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          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with entries",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Terms with French translations",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Category theory",
          "orig": "en:Category theory",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "text": "2010, Matthew Ando, Andrew J. Blumberg, David Gepner, Twists of K-Theory and TMF, Robert S. Doran, Greg Friedman, Jonathan Rosenberg, Superstrings, Geometry, Topology, and C*-algebras, American Mathematical Society, page 34,\nMoreover, f∗ admits both a left adjoint f_! and a right adjoint f_∗, given by left and right Kan extension along the map SingY→SingX, respectively. Note that this is left and right Kan extension in the ∞-categorical sense, which amounts to homotopy left and right Kan extension on the level of simplicial categories or model categories."
        },
        {
          "text": "2012, Rolf Hinze, Kan Extensions for Program Optimisation, Or: Art and Dan Explain an Old Trick, Jeremy Gibbons, Pablo Nogueira (editors), Mathematics of Program Construction: 11th International Conference, MPC 2012, Proceedings, Springer, Lecture Notes in Computer Science: 7342, page 336,\nWe can specialise Kan extensions to the preorder setting, if we equip a preorder with a monoidal structure: an associative operation that is monotone and that has a neutral element."
        },
        {
          "ref": "2013, Franz Vogler, Derived Manifolds from Functors of Points, Logos Verlag, page 5:",
          "text": "We are going to introduce the direct and inverse image functor for presheaves as special Kan extensions and show that they behave well with respect to global and local model structures on simplicial model categories.",
          "type": "quote"
        }
      ],
      "glosses": [
        "A construct that generalizes the notion of extending a function's domain of definition."
      ],
      "id": "en-Kan_extension-en-noun-0Si~8Y1q",
      "links": [
        [
          "category theory",
          "category theory"
        ],
        [
          "construct",
          "construct"
        ],
        [
          "generalize",
          "generalize"
        ],
        [
          "extend",
          "extend"
        ],
        [
          "function",
          "function"
        ],
        [
          "domain",
          "domain"
        ],
        [
          "definition",
          "definition"
        ]
      ],
      "raw_glosses": [
        "(category theory) A construct that generalizes the notion of extending a function's domain of definition."
      ],
      "topics": [
        "category-theory",
        "computing",
        "engineering",
        "mathematics",
        "natural-sciences",
        "physical-sciences",
        "sciences"
      ],
      "translations": [
        {
          "code": "fr",
          "lang": "French",
          "sense": "construct that generalizes the extending of a function's domain of definition",
          "tags": [
            "feminine"
          ],
          "word": "extension de Kan"
        }
      ],
      "wikipedia": [
        "Daniel Kan",
        "Kan extension"
      ]
    }
  ],
  "word": "Kan extension"
}

{
  "etymology_text": "Named after Jewish and Dutch mathematician Daniel M. Kan (1927–2013), who constructed certain (Kan) extensions using limits in 1960.",
  "forms": [
    {
      "form": "Kan extensions",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "Kan extension (plural Kan extensions)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English eponyms",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "Entries with translation boxes",
        "Pages with 1 entry",
        "Pages with entries",
        "Terms with French translations",
        "en:Category theory"
      ],
      "examples": [
        {
          "text": "2010, Matthew Ando, Andrew J. Blumberg, David Gepner, Twists of K-Theory and TMF, Robert S. Doran, Greg Friedman, Jonathan Rosenberg, Superstrings, Geometry, Topology, and C*-algebras, American Mathematical Society, page 34,\nMoreover, f∗ admits both a left adjoint f_! and a right adjoint f_∗, given by left and right Kan extension along the map SingY→SingX, respectively. Note that this is left and right Kan extension in the ∞-categorical sense, which amounts to homotopy left and right Kan extension on the level of simplicial categories or model categories."
        },
        {
          "text": "2012, Rolf Hinze, Kan Extensions for Program Optimisation, Or: Art and Dan Explain an Old Trick, Jeremy Gibbons, Pablo Nogueira (editors), Mathematics of Program Construction: 11th International Conference, MPC 2012, Proceedings, Springer, Lecture Notes in Computer Science: 7342, page 336,\nWe can specialise Kan extensions to the preorder setting, if we equip a preorder with a monoidal structure: an associative operation that is monotone and that has a neutral element."
        },
        {
          "ref": "2013, Franz Vogler, Derived Manifolds from Functors of Points, Logos Verlag, page 5:",
          "text": "We are going to introduce the direct and inverse image functor for presheaves as special Kan extensions and show that they behave well with respect to global and local model structures on simplicial model categories.",
          "type": "quote"
        }
      ],
      "glosses": [
        "A construct that generalizes the notion of extending a function's domain of definition."
      ],
      "links": [
        [
          "category theory",
          "category theory"
        ],
        [
          "construct",
          "construct"
        ],
        [
          "generalize",
          "generalize"
        ],
        [
          "extend",
          "extend"
        ],
        [
          "function",
          "function"
        ],
        [
          "domain",
          "domain"
        ],
        [
          "definition",
          "definition"
        ]
      ],
      "raw_glosses": [
        "(category theory) A construct that generalizes the notion of extending a function's domain of definition."
      ],
      "topics": [
        "category-theory",
        "computing",
        "engineering",
        "mathematics",
        "natural-sciences",
        "physical-sciences",
        "sciences"
      ],
      "wikipedia": [
        "Daniel Kan",
        "Kan extension"
      ]
    }
  ],
  "translations": [
    {
      "code": "fr",
      "lang": "French",
      "sense": "construct that generalizes the extending of a function's domain of definition",
      "tags": [
        "feminine"
      ],
      "word": "extension de Kan"
    }
  ],
  "word": "Kan extension"
}