Group object

In mathematics, group objects are certain generalizations of groups which 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.

Table of contents

1 Definition
2 Examples
3 Group theory generalized

Definition

Formally, we start with a category C which has a terminal object 1 and in which any two objects have a product. A group object in C is an object G of C together with morphisms

m : G × 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) are satisfied

m is associative, i.e. m(m × id_G) = m (id_G × m) as morphisms G × G × G → G; here we identify G × (G × G) in a canonical manner with (G × G) × G.
e is a two-sided unit of m, i.e. m (id_G × e) = p₁, where p₁ : G × 1 → G is the canonical projection, and m (e × id_G) = p₂, where p₂ : 1 × G → G is the canonical projection
inv 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 (also called the counit) with e, then m (id_G × inv) d = e_G and m (inv × id_G) d = e_G.

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 differentiable maps.
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 schemess.
The group objects in the category of groups (or monoids) are essentially 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.

Group theory generalized

Much of group theory can be formulated in the context of the more general group objects. The notions of group homomorphism, subgroup, normal subgroup and the isomorphism theorems are typical examples. However, results of group theory that talk about individual elements, or the order of specific elements or subgroups, normally cannot be generalized to group objects in a straight-forward manner.