group object

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group object

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Noun

group object (plural 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”, in 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 In the process, attention is withdrawn from individual elements and placed more strongly on operations.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