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.

Contents

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 × GG (thought of as the "group multiplication")
  • e : 1 → G (thought of as the "inclusion of the identity element")
  • inv: GG (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 × idG) = m (idG × m) as morphisms G × G × GG; 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 × GG is the canonical projection
  • inv is a two-sided inverse for m, i.e. if d : GG × G is the diagonal map, and eG : GG 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.

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. eG : GG 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: GG \oplus G, a "coidentity" e: G → 0, and a "coinversion" inv: GG, 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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