Group object

In category theory, a branch of mathematics, group objects are certain generalizations of groups that are built on more complicated structures than 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 continuous.

Definition
Formally, we start with a category C with finite products (i.e. C has a terminal object 1 and any two objects of C have a product). A group object in C is an object G of C together with morphisms such that the following properties (modeled on the group axioms – more precisely, on the definition of a group used in universal algebra) are satisfied
 * 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")
 * m is associative, i.e. m (m &times; idG) = m (idG &times; m) as morphisms G &times; G &times; G → G, and where e.g. m &times; idG : 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 (idG &times; e) = p1, where p1 : G &times; 1 → G is the canonical projection, and m (e &times; idG) = p2, where p2 : 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 eG : G → G is the composition of the unique morphism G → 1 (also called the counit) with e, then m (idG &times; inv) d = eG and m (inv &times; idG) d = eG.

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 structure (G, m, u, −1) can be defined can be considered 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 u of G, and the map inv assigns to every group element its inverse. eG : 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.
 * The strict 2-group is the group object in the category of small categories.
 * Given a category C with finite coproducts, a cogroup object is an object G of C together with a "comultiplication" m: G → G $$\oplus$$ G, a "coidentity" e: G → 0, and a "coinversion" inv: G → G that 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.