Editing Group object (section)

== Examples ==
* Each set ''G'' for which a [[group (mathematics)|group]] structure (''G'', ''m'', ''u'', <sup>−1</sup>) can be defined can be considered a group object in the category of [[set theory|sets]]. The map ''m'' is the group operation, the map ''e'' (whose domain is a [[singleton (mathematics)|singleton]]) picks out the identity element ''u'' of ''G'', and the map ''inv'' assigns to every group element its inverse. ''e''<sub>''G''</sub> : ''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 [[topology|topological spaces]] with [[continuous function (topology)|continuous functions]].
* A [[Lie group]] is a group object in the category of [[manifold|smooth manifolds]] with [[smooth map]]s.
* A [[Lie supergroup]] is a group object in the category of [[supermanifold]]s.
* An [[algebraic group]] is a group object in the category of [[algebraic variety|algebraic varieties]]. In modern [[algebraic geometry]], one considers the more general [[group scheme]]s, group objects in the category of [[scheme (mathematics)|scheme]]s.
* A localic group is a group object in the category of [[locale (mathematics)|locales]].
* The group objects in the category of groups (or [[monoid]]s) are the [[abelian group]]s. 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]].
* The strict [[2-group]] is the group object in the [[category of small categories]].
* Given a category ''C'' with finite [[coproduct]]s, a '''cogroup object''' is an object ''G'' of ''C'' together with a "comultiplication" ''m'': ''G'' → ''G'' <math>\oplus</math> ''G,'' a "coidentity" ''e'': ''G'' → 0, and a "coinversion" ''inv'': ''G'' → ''G'' that satisfy the [[dual (category theory)|dual]] versions of the axioms for group objects. Here 0 is the [[initial object]] of ''C''. Cogroup objects occur naturally in [[algebraic topology]].