group object

English

Noun

group object (plural 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 ${\\displaystyle H}$ 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 ${\\displaystyle {\\text{Grp}}(C)}$ .

Usage notes

Alternatively, and more concisely, an object ${\\displaystyle X\\in C}$ such that for any ${\\displaystyle Y\\in C}$ , the set of morphisms ${\\displaystyle {\\text{Hom}}_{C}(Y,X)}$ is a group and the correspondence ${\\displaystyle Y\\rightarrow {\\text{Hom}}_{C}(Y,X)}$ is a functor from ${\\displaystyle C}$ into the category of groups ${\\displaystyle {\\text{Gr}}}$ .

Group objects generalise the concept of group to objects of greater complexity than mere sets. In the process, attention is withdrawn from individual elements and placed more strongly on operations. A typical example of a group object might be a topological group where the object is a topological space on which the group operations are differentiable.

单词	group object
释义	group object English Noun group object (plural 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 ${\\displaystyle H}$ 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 ${\\displaystyle {\\text{Grp}}(C)}$ . Usage notes Alternatively, and more concisely, an object ${\\displaystyle X\\in C}$ such that for any ${\\displaystyle Y\\in C}$ , the set of morphisms ${\\displaystyle {\\text{Hom}}_{C}(Y,X)}$ is a group and the correspondence ${\\displaystyle Y\\rightarrow {\\text{Hom}}_{C}(Y,X)}$ is a functor from ${\\displaystyle C}$ into the category of groups ${\\displaystyle {\\text{Gr}}}$ . Group objects generalise the concept of group to objects of greater complexity than mere sets. In the process, attention is withdrawn from individual elements and placed more strongly on operations. A typical example of a group object might be a topological group where the object is a topological space on which the group operations are differentiable. Further reading Category theory on Wikipedia.Wikipedia Hom functor on Wikipedia.Wikipedia Universal algebra on Wikipedia.Wikipedia Group object on Encyclopedia of Mathematics group object on nLab
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group object

English

Noun

Usage notes

Further reading