|
|
| Line 1: |
Line 1: |
| In [[category theory]], a branch of [[mathematics]], '''group objects''' are certain generalizations of [[group (mathematics)|groups]] which are built on more complicated structures than [[Set (mathematics)|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 [[continuity (topology)|continuous]].
| | Hello, my name is Andrew and my wife doesn't like it at all. Some time ago he selected to reside in North Carolina and he doesn't strategy on altering it. The favorite pastime for him and his children is style and he'll be beginning something else along with it. Distributing manufacturing is exactly where my main income arrives from and it's some thing I really appreciate.<br><br>Take a look at my web-site: [http://browniepunto.com/?q=content/provide-little-lift-your-garden-advice clairvoyant psychic] |
| | |
| ==Definition==
| |
| | |
| Formally, we start with a [[category (mathematics)|category]] ''C'' with finite products (i.e. ''C'' has a [[terminal object]] 1 and any two objects of ''C'' have a [[product (category theory)|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 [[Universal algebra#Groups|definition of a group]] used in [[universal algebra]]) are satisfied
| |
| * ''m'' is associative, i.e. ''m''(''m'' × id<sub>''G''</sub>) = ''m'' (id<sub>''G''</sub> × ''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<sub>''G''</sub> × ''e'') = ''p''<sub>1</sub>, where ''p''<sub>1</sub> : ''G'' × 1 → ''G'' is the canonical projection, and ''m'' (''e'' × id<sub>''G''</sub>) = ''p''<sub>2</sub>, where ''p''<sub>2</sub> : 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''<sub>''G''</sub> : ''G'' → ''G'' is the composition of the unique morphism ''G'' → 1 (also called the counit) with ''e'', then ''m'' (id<sub>''G''</sub> × ''inv'') ''d'' = ''e''<sub>''G''</sub> and ''m'' (''inv'' × id<sub>''G''</sub>) ''d'' = ''e''<sub>''G''</sub>.
| |
| | |
| 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.
| |
| | |
| == Examples ==
| |
| * A [[group (mathematics)|group]] can be viewed as a group object in the category of [[set theory|sets]]. The map ''m'' is the group operation, the map ''e'' (whose domain is a [[singleton (mathematics)|singleton]]) picks out the identity element of the group, and the map ''inv'' assigns to every group element its inverse. ''e''<sub>''G''</sub> : ''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 [[topology|topological spaces]] with [[continuous function (topology)|continuous functions]].
| |
| * A [[Lie group]] is a group object in the category of [[manifold|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 variety|algebraic varieties]]. In modern [[algebraic geometry]], one considers the more general [[group scheme]]s, group objects in the category of [[scheme (mathematics)|scheme]]s.
| |
| * A [[localic group]] is a group object in the category of [[locale (mathematics)|locales]].
| |
| * The group objects in the category of groups (or [[monoid]]s) are 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. See also [[Eckmann-Hilton argument]].
| |
| * Given a category ''C'' with finite [[coproduct]]s, a '''cogroup object''' is an object ''G'' of ''C'' together with a "comultiplication" ''m'': ''G'' → ''G'' <math>\oplus</math> ''G,'' a "coidentity" ''e'': ''G'' → 0, and a "coinversion" ''inv'': ''G'' → ''G'', which satisfy the [[dual (category theory)|dual]] versions of the axioms for group objects. Here 0 is the [[initial object]] of ''C''. Cogroup objects occur naturally in [[algebraic topology]].
| |
| | |
| ==Group theory generalized== | |
| | |
| 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.{{Fact|date=December 2007}} 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 straightforward manner.{{Fact|date=July 2011}}
| |
| | |
| ==See also==
| |
| | |
| * [[Hopf algebra]]s can be seen as a generalization of group objects to [[monoidal category|monoidal categories]].
| |
| | |
| ==References==
| |
| {{Refimprove|date=December 2007}}
| |
| * {{Lang Algebra|edition=3r}}
| |
| | |
| [[Category:Group theory]]
| |
| [[Category:Objects (category theory)]]
| |
Hello, my name is Andrew and my wife doesn't like it at all. Some time ago he selected to reside in North Carolina and he doesn't strategy on altering it. The favorite pastime for him and his children is style and he'll be beginning something else along with it. Distributing manufacturing is exactly where my main income arrives from and it's some thing I really appreciate.
Take a look at my web-site: clairvoyant psychic