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In category theory, a branch of mathematics, group objects are certain generalizations of groups which 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== 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 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Group object」の詳細全文を読む スポンサード リンク
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