"direct limit" meaning in All languages combined

{
  "antonyms": [
    {
      "word": "inverse limit"
    }
  ],
  "forms": [
    {
      "form": "direct limits",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "direct limit (plural direct limits)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
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          "source": "w"
        },
        {
          "_dis": "71 29",
          "kind": "other",
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          "parents": [],
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      ],
      "examples": [
        {
          "text": "A direct limit has “canonical functions” which map each element of the disjoint union to its equivalence class.",
          "type": "example"
        },
        {
          "text": "Direct limits in the algebraic sense are models of category-theoretic colimits.",
          "type": "example"
        }
      ],
      "glosses": [
        "A set of equivalence classes which partition the disjoint union of the members of a direct system; each equivalence class being a sort of “drainage basin” of the mappings (of the morphisms) of the direct system, if these are analogically considered as “rivers”. (If i<k,j<k in the indexing poset, then there exist f_ik:A_i→A_k and f_jk:A_j→A_k. If a_i∈A_i,a_j∈A_j such that f_ik(a_i)=f_jk(a_j) then a_i∼a_j. If k = j then f_jj(a_j)=a_j,f_ij(a_i)=a_j.)"
      ],
      "id": "en-direct_limit-en-noun-vf9zt9iZ",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "equivalence class",
          "equivalence class"
        ],
        [
          "disjoint union",
          "disjoint union"
        ],
        [
          "direct system",
          "direct system"
        ],
        [
          "drainage basin",
          "drainage basin"
        ]
      ],
      "raw_glosses": [
        "(algebra) A set of equivalence classes which partition the disjoint union of the members of a direct system; each equivalence class being a sort of “drainage basin” of the mappings (of the morphisms) of the direct system, if these are analogically considered as “rivers”. (If i<k,j<k in the indexing poset, then there exist f_ik:A_i→A_k and f_jk:A_j→A_k. If a_i∈A_i,a_j∈A_j such that f_ik(a_i)=f_jk(a_j) then a_i∼a_j. If k = j then f_jj(a_j)=a_j,f_ij(a_i)=a_j.)"
      ],
      "related": [
        {
          "_dis1": "100 0",
          "word": "direct system"
        }
      ],
      "synonyms": [
        {
          "_dis1": "100 0",
          "word": "inductive limit"
        }
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ]
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        }
      ],
      "glosses": [
        "a colimit"
      ],
      "id": "en-direct_limit-en-noun-CX01TUZA",
      "links": [
        [
          "category theory",
          "category theory"
        ],
        [
          "colimit",
          "colimit"
        ]
      ],
      "raw_glosses": [
        "(category theory) a colimit"
      ],
      "topics": [
        "category-theory",
        "computing",
        "engineering",
        "mathematics",
        "natural-sciences",
        "physical-sciences",
        "sciences"
      ]
    }
  ],
  "wikipedia": [
    "direct limit"
  ],
  "word": "direct limit"
}

{
  "antonyms": [
    {
      "word": "inverse limit"
    }
  ],
  "categories": [
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    "English entries with incorrect language header",
    "English lemmas",
    "English multiword terms",
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    "Pages with 1 entry",
    "Pages with entries"
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  "forms": [
    {
      "form": "direct limits",
      "tags": [
        "plural"
      ]
    }
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  "head_templates": [
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      "expansion": "direct limit (plural direct limits)",
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  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "direct system"
    }
  ],
  "senses": [
    {
      "categories": [
        "English terms with usage examples",
        "en:Algebra"
      ],
      "examples": [
        {
          "text": "A direct limit has “canonical functions” which map each element of the disjoint union to its equivalence class.",
          "type": "example"
        },
        {
          "text": "Direct limits in the algebraic sense are models of category-theoretic colimits.",
          "type": "example"
        }
      ],
      "glosses": [
        "A set of equivalence classes which partition the disjoint union of the members of a direct system; each equivalence class being a sort of “drainage basin” of the mappings (of the morphisms) of the direct system, if these are analogically considered as “rivers”. (If i<k,j<k in the indexing poset, then there exist f_ik:A_i→A_k and f_jk:A_j→A_k. If a_i∈A_i,a_j∈A_j such that f_ik(a_i)=f_jk(a_j) then a_i∼a_j. If k = j then f_jj(a_j)=a_j,f_ij(a_i)=a_j.)"
      ],
      "links": [
        [
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        ],
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          "disjoint union"
        ],
        [
          "direct system",
          "direct system"
        ],
        [
          "drainage basin",
          "drainage basin"
        ]
      ],
      "raw_glosses": [
        "(algebra) A set of equivalence classes which partition the disjoint union of the members of a direct system; each equivalence class being a sort of “drainage basin” of the mappings (of the morphisms) of the direct system, if these are analogically considered as “rivers”. (If i<k,j<k in the indexing poset, then there exist f_ik:A_i→A_k and f_jk:A_j→A_k. If a_i∈A_i,a_j∈A_j such that f_ik(a_i)=f_jk(a_j) then a_i∼a_j. If k = j then f_jj(a_j)=a_j,f_ij(a_i)=a_j.)"
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ]
    },
    {
      "categories": [
        "en:Category theory"
      ],
      "glosses": [
        "a colimit"
      ],
      "links": [
        [
          "category theory",
          "category theory"
        ],
        [
          "colimit",
          "colimit"
        ]
      ],
      "raw_glosses": [
        "(category theory) a colimit"
      ],
      "topics": [
        "category-theory",
        "computing",
        "engineering",
        "mathematics",
        "natural-sciences",
        "physical-sciences",
        "sciences"
      ]
    }
  ],
  "synonyms": [
    {
      "word": "inductive limit"
    }
  ],
  "wikipedia": [
    "direct limit"
  ],
  "word": "direct limit"
}