"inverse limit" meaning in All languages combined

{
  "antonyms": [
    {
      "word": "direct limit"
    }
  ],
  "forms": [
    {
      "form": "inverse limits",
      "tags": [
        "plural"
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      "expansion": "inverse limit (plural inverse limits)",
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
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          "kind": "topical",
          "langcode": "en",
          "name": "Algebra",
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      "examples": [
        {
          "text": "An inverse limit has “natural projections” which are restrictions of the projections of the Cartesian product (to a domain which is the inverse limit). The reason why the projections are described as “natural” would be the following: besides the functor from an index poset to the inverse system, there is another functor from the same index poset to the inverse limit of that system, this functor being a constant functor. Then there is a natural transformation from the constant functor to the inverse limit’s functor: the components of such natural transformation are the said “natural projections”.",
          "type": "example"
        },
        {
          "text": "Inverse limits are concrete-categorical versions of limits.",
          "type": "example"
        }
      ],
      "glosses": [
        "A subset of the Cartesian product of all the members of an inverse system, such that a member M of the subset is a sort of “cross section” of the inverse system (as fiber bundle) induced by the morphisms of it. (If i<j in the indexing poset then f_ij:A_j→A_i in the inverse system and if a_i∈A_i, a_j∈A_j are components of M then f_ij(a_j)=a_i)."
      ],
      "id": "en-inverse_limit-en-noun--ba0uPUO",
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          "inverse system",
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          "fiber bundle",
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      "raw_glosses": [
        "(algebra) A subset of the Cartesian product of all the members of an inverse system, such that a member M of the subset is a sort of “cross section” of the inverse system (as fiber bundle) induced by the morphisms of it. (If i<j in the indexing poset then f_ij:A_j→A_i in the inverse system and if a_i∈A_i, a_j∈A_j are components of M then f_ij(a_j)=a_i)."
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        "(category theory) a limit"
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          "word": "inverse system"
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        {
          "_dis1": "38 62",
          "word": "projective limit"
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      "examples": [
        {
          "text": "An inverse limit has “natural projections” which are restrictions of the projections of the Cartesian product (to a domain which is the inverse limit). The reason why the projections are described as “natural” would be the following: besides the functor from an index poset to the inverse system, there is another functor from the same index poset to the inverse limit of that system, this functor being a constant functor. Then there is a natural transformation from the constant functor to the inverse limit’s functor: the components of such natural transformation are the said “natural projections”.",
          "type": "example"
        },
        {
          "text": "Inverse limits are concrete-categorical versions of limits.",
          "type": "example"
        }
      ],
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        "A subset of the Cartesian product of all the members of an inverse system, such that a member M of the subset is a sort of “cross section” of the inverse system (as fiber bundle) induced by the morphisms of it. (If i<j in the indexing poset then f_ij:A_j→A_i in the inverse system and if a_i∈A_i, a_j∈A_j are components of M then f_ij(a_j)=a_i)."
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        "(algebra) A subset of the Cartesian product of all the members of an inverse system, such that a member M of the subset is a sort of “cross section” of the inverse system (as fiber bundle) induced by the morphisms of it. (If i<j in the indexing poset then f_ij:A_j→A_i in the inverse system and if a_i∈A_i, a_j∈A_j are components of M then f_ij(a_j)=a_i)."
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  "wikipedia": [
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  "word": "inverse limit"
}