Group object explained
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
- 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 – more precisely, on the definition of a group used in universal algebra) are satisfied
- m is associative, i.e. m (m × idG) = m (idG × m) as morphisms G × G × G → G, and where e.g. m × idG : G × 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 (idG × e) = p1, where p1 : G × 1 → G is the canonical projection, and m (e × idG) = 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 eG : G → G is the composition of the unique morphism G → 1 (also called the counit) with e, then m (idG × inv) d = eG and m (inv × 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
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.
See also
