# Group object

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[edit]

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*× id_{G}) =*m*(id_{G}×*m*) as morphisms*G*×*G*×*G*→*G*, and where e.g.*m*× id_{G}:*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*(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 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}.

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[edit]

- 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.*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 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[edit]

- Hopf algebras can be seen as a generalization of group objects to monoidal categories.
- Groupoid object

## References[edit]

- Awodey, Steve (2010),
*Category Theory*, Oxford University Press, ISBN 9780199587360 - Lang, Serge (2002),
*Algebra*, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001