English Dictionary

group object

名詞

(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.

日本語の意味

圏論において、与えられた圏Cの中の対象Xで、群論における二項演算（掛け算）、単位元、逆元に対応する射が定義され、これらの射が結合性や逆元、単位元の存在に対応する性質を満たすような対象を指す。

In the category of schemes, a group object is an object X equipped with morphisms for multiplication, identity, and inverse that satisfy the group axioms.

スキームの圏では、群対象とは乗法、単位元、逆元に対応する射が与えられ、それらが群の公理を満たす対象Xのことである。

意味（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.

If H is another group object in C, a morphism f:G→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. Thus we form the category textit Gp(C) of all group objects in C; one important example is textit Gp( textit Ho).

group objects （ plural ）