group object

plural group objects

1. category theoryGiven 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.

   2 more examples

   If H is another group object in C, a morphism f#58;G#92;rightarrowH 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 #92;mu. Thus we form the category #92;textit#123;Gp#125;(C) of all group objects in C; one important example is #92;textit#123;Gp#125;(#92;textit#123;Ho#125;).

   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

   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).