group object - Weblio 英和・和英辞典 (original) (raw)

単語を追加

× この辞書を今後表示しない

※辞書の非表示は、設定画面から変更可能

group object

名詞

group object (複数形 group objects)

(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.
- 1995, J. Michael Boardman, Chapter 14: Stable Operations in Generalized Cohomology, I.M. James (editor), Handbook of Algebraic Topology, Elsevier (North-Holland), page 617,
  If H {\displaystyle H} $H$ is another group object in C {\displaystyle C} $C$ , a morphism f : G → H {\displaystyle f:G\rightarrow H} $f:G\rightarrow H$ 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 μ {\displaystyle \mu } $\mu$ . Thus we form the category Gp ( C ) {\displaystyle {\textit {Gp}}(C)} ${\textit {Gp}}(C)$ of all group objects in C {\displaystyle C} $C$ ; one important example is Gp ( Ho ) {\displaystyle {\textit {Gp}}({\textit {Ho}})} ${\textit {Gp}}({\textit {Ho}})$ .
- 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,
  The 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 ) {\displaystyle {\text{Grp}}(C)} ${\text{Grp}}(C)$ .