Group object
of a category
An object of a category
such that for any
the set of morphisms
is a group, while the correspondence
is a functor from
into the category of groups Gr. A homomorphism of a group object
into a group object
is a morphism
of
such that for any
the corresponding mapping
is a homomorphism of groups. The group objects of a category
and homomorphisms between them form the category
. The functor
establishes an equivalence between the category
and the category of representable pre-sheaves of groups on
. If the values of the functor
belong to the subcategory Ab of Abelian groups, then the group object
is said to be commutative or Abelian. If
has finite products and a final object
, a group object
of
is defined by the following properties.
There exist morphisms (multiplication),
(inversion) and
(a unit) satisfying the following axioms.
Associativity. The diagram
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is commutative.
Existence of a unit element. The diagram
![]() |
is commutative.
Existence of an inverse element. The diagram
![]() |
is commutative. Here is the canonical morphism of
into the final object
, while
is the diagonal morphism.
If is the category of sets Ens, group objects are precisely groups. The final object of the category Ens is the set
consisting of the single element
. Axiom a) denotes the associativity of the binary operation given by the morphism
. The morphism
is the mapping of inversion, while the morphism
is the mapping of the set
into
, whose image is equal to the unit element in
.
In a similar manner it is possible to define a ring object of a category and, generally, to specify an algebraic structure on an object of a category [2].
References
[1] | Yu.I. Manin, "The theory of commutative formal groups over fields of finite characteristic" Russian Math. Surveys , 18 (1963) pp. 1–80 Uspekhi Mat. Nauk , 18 : 6 (1963) pp. 3–90 |
[2] | M. Demazure, A. Grothendieck, "Schémas en groupes I" , Lect. notes in math. , 151–153 , Springer (1970) |
Comments
Group objects, in particular categories, are often objects of interest in their own right. For example, topological groups (cf. Topological group) are group objects in the category of topological spaces and continuous mappings; Lie groups (cf. Lie group) are group objects in the category of smooth manifolds; and sheaves of groups on a given space are group objects in the category of sheaves of sets on
. A group object in a category of the form
is an object of
equipped with two commuting group structures; it is easily seen that in this case the two structures must coincide and be Abelian, and conversely an Abelian group structure commutes with itself, so that
is isomorphic to the category
of Abelian group objects in
. A functor which preserves finite products (including the final object) preserves group objects; using this and the above identification, one obtains an easy proof of the result that the fundamental group of a topological group is Abelian.
References
[a1] | S. MacLane, "Categories for the working mathematician" , Springer (1971) pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7 |
[a2] | B. Eckmann, P.J. Hilton, "Group-like structures in general categories I. Multiplications and comultiplications" Math. Ann. , 145 (1962) pp. 227–255 |
Group object. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Group_object&oldid=18656