Group Object - Examples

Examples

• A group can be viewed as a group object in the category of sets. The map m is the group operation, the map e (whose domain is a singleton) picks out the identity element of the group, and the map inv assigns to every group element its inverse. e_G : G → G is the map that sends every element of G to the identity element.
• A topological group is a group object in the category of topological spaces with continuous functions.
• A Lie group is a group object in the category of smooth manifolds with smooth maps.
• A Lie supergroup is a group object in the category of supermanifolds.
• An algebraic group is a group object in the category of algebraic varieties. In modern algebraic geometry, one considers the more general group schemes, group objects in the category of schemes.
• A localic group is a group object in the category of locales.
• The group objects in the category of groups (or monoids) are the Abelian groups. The reason for this is that, if inv is assumed to be a homomorphism, then G must be abelian. More precisely: if A is an abelian group and we denote by m the group multiplication of A, by e the inclusion of the identity element, and by inv the inversion operation on A, then (A,m,e,inv) is a group object in the category of groups (or monoids). Conversely, if (A,m,e,inv) is a group object in one of those categories, then m necessarily coincides with the given operation on A, e is the inclusion of the given identity element on A, inv is the inversion operation and A with the given operation is an abelian group. See also Eckmann-Hilton argument.
• Given a category C with finite coproducts, a cogroup object is an object G of C together with a "comultiplication" m: G → G G, a "coidentity" e: G → 0, and a "coinversion" inv: G → G, which satisfy the dual versions of the axioms for group objects. Here 0 is the initial object of C. Cogroup objects occur naturally in algebraic topology.