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 × idG) = m (idG × 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 (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 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.
Read more about this topic: Group Object
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