Group object

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In category theory, a branch of 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.

• 1 Definition
• 2 Examples
• 3 Group theory generalized
• 4 See also
• 5 References

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) 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.

A more general definition is that G is a group object in any 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). The two definitions are equivalent if C has finite products and a terminal object.

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 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 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. See also Eckmann-Hilton argument.
• 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, which 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.

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