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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 × 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 twosided unit of m, i.e. m (id_{G} × e) = p_{1}, where p_{1} : G × 1 → G is the canonical projection, and m (e × id_{G}) = p_{2}, where p_{2} : 1 × G → G is the canonical projection
 inv is a twosided 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 EckmannHilton 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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