Group object

In , 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. Formally, we start with a C with finite products (i.e. C has a terminal object 1 and any two of C have a ). 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 . 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.