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 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 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''.
- 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 Map s.
- A Lie Supergroup is a group object in the category of Supermanifold s.
- An Algebraic Group is a group object in the category of Algebraic Varieties . In modern Algebraic Geometry , one considers the more general Group Scheme s, group objects in the category of Scheme s.
- The group objects in the category of groups (or Monoid s) are essentially the Abelian Group s. 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.
Much of Group Theory can be formulated in the context of the more general group objects. The notions of Group Homomorphism , Subgroup , Normal Subgroup and the Isomorphism Theorem s are typical examples. However, results of group theory that talk about individual elements, or the order of specific elements or subgroups, normally cannot be generalized to group objects in a straight-forward manner.
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