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. A group object in C is an object G of C together with morphisms

m : G × G → G
e : 1 → G
inv : G → G

such that the following properties are satisfiedm is associative, i.e. m = m as morphisms G × G × G → G, and where e.g. m × id_G : G × G × G → G × G; here we identify G × in a canonical manner with × G. e is a two-sided unit of m, i.e. m = p₁, where p₁ : G × 1 → G is the canonical projection, and m = p₂, where p₂ : 1 × G → G is the canonical projectioninv is a two-sided inverse for m, i.e. if d : G → G × G is the diagonal map, and e_G : G → G is the composition of the unique morphism G → 1 with e, then m ''d = e''_G and m ''d = e''_G.
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 from X to G such that the association of X to Hom is a functor from C to the category of groups.
Yet another way to state the above is to define a group object as a monoid object in the cartesian monoidal category, together with an inverse morphism satisfying the above conditions.

Examples

Each set G for which a group structure can be defined can be considered a group object in the category of sets. The map m is the group operation, the map e picks out the identity element u of G, 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 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 is a group object in the category of groups. Conversely, if 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 ''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.