group objectとは 意味・読み方・使い方

名詞

group object (複数形 group objects)

1. (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 ${\displaystyle H}$ is another group object in ${\displaystyle C}$, a morphism ${\displaystyle f:G\rightarrow H}$ 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 ${\displaystyle \mu }$. Thus we form the category ${\displaystyle {\textit {Gp}}(C)}$ of all group objects in ${\displaystyle C}$; one important example is ${\displaystyle {\textit {Gp}}({\textit {Ho}})}$.

   • 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 ${\displaystyle {\text{Grp}}(C)}$.