Editing Group object

In [[category theory]], a branch of [[mathematics]], '''group objects''' are certain generalizations of [[group (mathematics)|groups]] that are built on more complicated structures than [[Set (mathematics)|sets]]. A typical example of a group object is a [[topological group]], a group whose underlying set is a [[topological space]] such that the group operations are [[continuity (topology)|continuous]].

==Definition==

Formally, we start with a [[category (mathematics)|category]] ''C'' with finite products (i.e. ''C'' has a [[terminal object]] 1 and any two objects of ''C'' have a [[product (category theory)|product]]). A '''group object''' in ''C'' is an object ''G'' of ''C'' together with [[morphism]]s
*''m'' : ''G'' &times; ''G'' → ''G'' (thought of as the "group multiplication")
*''e'' : 1 → ''G'' (thought of as the "inclusion of the identity element")
*''inv'' : ''G'' → ''G'' (thought of as the "inversion operation")
such that the following properties (modeled on the group axioms –  more precisely, on the [[Universal algebra#Groups|definition of a group]] used in [[universal algebra]]) are satisfied
* ''m'' is associative, i.e. ''m'' (''m'' &times; id<sub>''G''</sub>) = ''m'' (id<sub>''G''</sub> &times; ''m'') as morphisms ''G'' &times; ''G'' &times; ''G'' → ''G'', and where e.g. ''m'' &times; id<sub>''G''</sub> : ''G'' &times; ''G'' &times; ''G'' → ''G'' &times; ''G''; here we identify ''G'' &times; (''G'' &times; ''G'') in a canonical manner with (''G'' &times; ''G'') &times; ''G''. 
* ''e'' is a two-sided unit of ''m'', i.e. ''m'' (id<sub>''G''</sub> &times; ''e'') = ''p''<sub>1</sub>, where ''p''<sub>1</sub> : ''G'' &times; 1 → ''G'' is the canonical projection, and  ''m'' (''e'' &times; id<sub>''G''</sub>) = ''p''<sub>2</sub>, where ''p''<sub>2</sub> : 1 &times; ''G'' → ''G'' is the canonical projection
* ''inv'' is a two-sided inverse for ''m'', i.e. if ''d'' : ''G'' → ''G'' &times; ''G'' is the diagonal map, and ''e''<sub>''G''</sub> : ''G'' → ''G'' is the composition of the unique morphism ''G'' → 1 (also called the counit) with ''e'', then ''m'' (id<sub>''G''</sub> &times; ''inv'') ''d'' = ''e''<sub>''G''</sub> and ''m'' (''inv'' &times; id<sub>''G''</sub>) ''d'' = ''e''<sub>''G''</sub>.

Note that this is stated in terms of maps – product and inverse must be maps in the category – and without any reference to underlying "elements" of the group object – categories in general do not have elements of their objects.

Another way to state the above is to say ''G'' is a group object in a category ''C'' if for every object ''X'' in ''C'', there is a group structure on the morphisms Hom(''X'', ''G'') from ''X'' to ''G'' such that the association of ''X'' to Hom(''X'', ''G'') is a (contravariant) [[functor]] from ''C'' to the [[category of groups]].

== 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]].

==See also==
* [[Hopf algebra]]s can be seen as a generalization of group objects to [[monoidal category|monoidal categories]].
*[[Groupoid object]]
*[[internal category]]<!-- a group object is an instance of an internal category. -->

==References==
{{Reflist}}
* {{Citation|title=Category Theory| last=Awodey|first=Steve|isbn=9780199587360|year=2010|publisher=Oxford University Press}}
* {{Lang Algebra|edition=3r}}

[[Category:Group theory]]
[[Category:Objects (category theory)]]