In
category theory, a branch of
mathematics, group objects are certain generalizations of
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
that are built on more complicated structures than
sets. A typical example of a group object is a
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
, a group whose underlying set is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
such that the group operations are
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
.
Definition
Formally, we start with a
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
''C'' with finite products (i.e. ''C'' has a
terminal object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element): ...
1 and any two
objects of ''C'' have a
product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
). 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
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular groups as the object of stu ...
) 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 the group object – categories in general do not have elements of 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
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
from ''C'' to the
category of groups
In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory.
Relation to other categories
There a ...
.
Examples
* Each set ''G'' for which a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
structure (''G'', ''m'', ''u'',
−1) can be defined can be considered a group object in the category of
sets. The map ''m'' is the group operation, the map ''e'' (whose domain is a
singleton
Singleton may refer to:
Sciences, technology Mathematics
* Singleton (mathematics), a set with exactly one element
* Singleton field, used in conformal field theory Computing
* Singleton pattern, a design pattern that allows only one instance ...
) picks out the identity element ''u'' of ''G'', 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
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
is a group object in the category of
topological spaces
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...
with
continuous functions
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
.
* A
Lie group is a group object in the category of
smooth manifolds
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
with
smooth map
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
s.
* A
Lie supergroup
The concept of supergroup is a generalization of that of group. In other words, every supergroup carries a natural group structure, but there may be more than one way to structure a given group as a supergroup. A supergroup is like a Lie group in t ...
is a group object in the category of
supermanifold
In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below.
Informal definition
An informal definition is com ...
s.
* An
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Ma ...
is a group object in the category of
algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
. In modern
algebraic geometry, one considers the more general
group scheme
In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have ...
s, group objects in the category of
schemes.
* A localic group is a group object in the category of
locales.
* The group objects in the category of groups (or
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoid ...
s) are the
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
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.
* The strict
2-group
In mathematics, a 2-group, or 2-dimensional higher group, is a certain combination of group and groupoid. The 2-groups are part of a larger hierarchy of ''n''-groups. In some of the literature, 2-groups are also called gr-categories or groupal ...
is the group object in the
category of small categories
In mathematics, specifically in category theory, the category of small categories, denoted by Cat, is the category whose objects are all small categories and whose morphisms are functors between categories. Cat may actually be regarded as a 2-cat ...
.
* Given a category ''C'' with finite
coproduct
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduc ...
s, a cogroup object is an object ''G'' of ''C'' together with a "comultiplication" ''m'': ''G'' → ''G''
''G,'' a "coidentity" ''e'': ''G'' → 0, and a "coinversion" ''inv'': ''G'' → ''G'' that satisfy the
dual versions of the axioms for group objects. Here 0 is the
initial object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element): ...
of ''C''. Cogroup objects occur naturally in
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
.
Group theory generalized
Much of
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
can be formulated in the context of the more general group objects. The notions of
group homomorphism
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
w ...
,
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
,
normal subgroup and the
isomorphism theorem
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist ...
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 straightforward manner.
See also
*
Hopf algebras can be seen as a generalization of group objects to
monoidal categories
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an object ''I'' that is both a left and r ...
.
*
Groupoid object
References
*
* {{Lang Algebra, edition=3r
Group theory
Objects (category theory)