Group Object - Definition

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 – 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; 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.

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 group – categories in general do not have elements to 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.