"group object" meaning in English

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        "plural"
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
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          "langcode": "en",
          "name": "Category theory",
          "orig": "en:Category theory",
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      "examples": [
        {
          "ref": "1995, J. Michael Boardman, “Chapter 14: Stable Operations in Generalized Cohomology”, in I.M. James, editor, Handbook of Algebraic Topology, Elsevier (North-Holland), page 617",
          "text": "If H is another group object in C, a morphism f#x3A;G#x5C;rightarrowH is a morphism of group objects if it commutes with the three structure morphisms; as is standard for sets and true generally (again by Lemma 7.7), it is enough to check #x5C;mu. Thus we form the category #x5C;textit#x7B;Gp#x7D;(C) of all group objects in C; one important example is #x5C;textit#x7B;Gp#x7D;(#x5C;textit#x7B;Ho#x7D;).",
          "type": "quotation"
        },
        {
          "text": "2005, Angelo Vistoli, Part 1: Grothendieck typologies, fibered categories, and descent theory, Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, Angelo Vistoli, Fundamental Algebraic Geometry: Grothendieck's FGA Explained, American Mathematical Society, page 20,\nThe identity is obviously a homomorphism from a group object to itself. Furthermore, the composite of homomorphisms of group objects is still a homomorphism; thus, group objects in a fixed category form a category, which we denote by Grp(C)."
        }
      ],
      "glosses": [
        "Given a category C, any object X ∈ C on which morphisms are defined corresponding to the group theoretic concepts of a binary operation (called multiplication), identity and inverse, such that multiplication is associative and properties are satisfied that correspond to the existence of inverse elements and the identity element."
      ],
      "id": "en-group_object-en-noun-hlhTNlj4",
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          "category theory",
          "category theory"
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          "category",
          "category"
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        [
          "object",
          "object"
        ],
        [
          "morphism",
          "morphism"
        ],
        [
          "group theoretic",
          "group theory"
        ],
        [
          "binary operation",
          "binary operation"
        ],
        [
          "identity",
          "identity"
        ],
        [
          "inverse",
          "inverse"
        ],
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          "associative",
          "associative"
        ]
      ],
      "raw_glosses": [
        "(category theory) Given a category C, any object X ∈ C on which morphisms are defined corresponding to the group theoretic concepts of a binary operation (called multiplication), identity and inverse, such that multiplication is associative and properties are satisfied that correspond to the existence of inverse elements and the identity element."
      ],
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        "category-theory",
        "computing",
        "engineering",
        "mathematics",
        "natural-sciences",
        "physical-sciences",
        "sciences"
      ],
      "wikipedia": [
        "group object"
      ]
    }
  ],
  "word": "group object"
}

{
  "forms": [
    {
      "form": "group objects",
      "tags": [
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  "lang_code": "en",
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      "examples": [
        {
          "ref": "1995, J. Michael Boardman, “Chapter 14: Stable Operations in Generalized Cohomology”, in I.M. James, editor, Handbook of Algebraic Topology, Elsevier (North-Holland), page 617",
          "text": "If H is another group object in C, a morphism f#x3A;G#x5C;rightarrowH is a morphism of group objects if it commutes with the three structure morphisms; as is standard for sets and true generally (again by Lemma 7.7), it is enough to check #x5C;mu. Thus we form the category #x5C;textit#x7B;Gp#x7D;(C) of all group objects in C; one important example is #x5C;textit#x7B;Gp#x7D;(#x5C;textit#x7B;Ho#x7D;).",
          "type": "quotation"
        },
        {
          "text": "2005, Angelo Vistoli, Part 1: Grothendieck typologies, fibered categories, and descent theory, Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, Angelo Vistoli, Fundamental Algebraic Geometry: Grothendieck's FGA Explained, American Mathematical Society, page 20,\nThe identity is obviously a homomorphism from a group object to itself. Furthermore, the composite of homomorphisms of group objects is still a homomorphism; thus, group objects in a fixed category form a category, which we denote by Grp(C)."
        }
      ],
      "glosses": [
        "Given a category C, any object X ∈ C on which morphisms are defined corresponding to the group theoretic concepts of a binary operation (called multiplication), identity and inverse, such that multiplication is associative and properties are satisfied that correspond to the existence of inverse elements and the identity element."
      ],
      "links": [
        [
          "category theory",
          "category theory"
        ],
        [
          "category",
          "category"
        ],
        [
          "object",
          "object"
        ],
        [
          "morphism",
          "morphism"
        ],
        [
          "group theoretic",
          "group theory"
        ],
        [
          "binary operation",
          "binary operation"
        ],
        [
          "identity",
          "identity"
        ],
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          "inverse",
          "inverse"
        ],
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          "associative",
          "associative"
        ]
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        "(category theory) Given a category C, any object X ∈ C on which morphisms are defined corresponding to the group theoretic concepts of a binary operation (called multiplication), identity and inverse, such that multiplication is associative and properties are satisfied that correspond to the existence of inverse elements and the identity element."
      ],
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      ],
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        "group object"
      ]
    }
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}