View source for Group object

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''of a category''

An object  $  X $
of a [[Category|category]]  $  C $
such that for any  $  Y \in  \mathop{\rm Ob} ( C) $
the set of morphisms  $   \mathop{\rm Hom} _ {C} ( Y, X) $
is a group, while the correspondence  $  Y \rightarrow  \mathop{\rm Hom} _ {C} ( Y, X) $
is a functor from  $  C $
into the category of groups Gr. A homomorphism of a group object  $  X $
into a group object  $  X _ {1} $
is a morphism  $  f:  X \rightarrow X _ {1} $
of  $  C $
such that for any  $  Y \in  \mathop{\rm Ob} ( C) $
the corresponding mapping  $   \mathop{\rm Hom} _ {C} ( Y, X) \rightarrow  \mathop{\rm Hom} _ {C} ( Y, X _ {1} ) $
is a homomorphism of groups. The group objects of a category  $  C $
and homomorphisms between them form the category  $   \mathop{\rm Gr} - C $. 
The functor  $  X \rightarrow h _ {X} =  \mathop{\rm Hom} _ {C} ( \cdot , X) $
establishes an equivalence between the category  $   \mathop{\rm Gr} - C $
and the category of representable pre-sheaves of groups on  $  C $. 
If the values of the functor  $  h _ {X} :  Y \rightarrow  \mathop{\rm Hom} _ {C} ( Y, X) $
belong to the subcategory Ab of Abelian groups, then the group object  $  X $
is said to be commutative or Abelian. If  $  C $
has finite products and a final object  $  e $, 
a group object  $  X $
of  $  C $
is defined by the following properties.

There exist morphisms  $  m:  X \times X \rightarrow X $(
multiplication),  $  r:  X \rightarrow X $(
inversion) and  $  \beta :  e \rightarrow X $(
a unit) satisfying the following axioms.

Associativity. The diagram

$$ 

\begin{array}{rcr}
X \times X \times X  &  \mathop \rightarrow \limits ^ { {m \times id }}   X \times X  \rightarrow ^ { m }    & X  \\
{} _ {id \times m }  \searrow  &{}  &\nearrow _ {id \times m }   \\
{}  &X \times X  &{}  \\
\end{array}

$$

is commutative.

Existence of a unit element. The diagram

$$ 

\begin{array}{rcr}
X \times X  & \rightarrow ^ { {p _ x} \times id } \ 
e \times X   \mathop \rightarrow \limits ^ { {\beta \times id }}    &X \times X  \\
{} _  \Delta  \nwarrow  &{}  &\swarrow _ {m}  \\
{}  &X \ 
\rightarrow _ { id }  X  &{}  \\
\end{array}

$$

is commutative.

Existence of an inverse element. The diagram

$$ 

\begin{array}{rcr}
X \times X  &  \mathop \rightarrow \limits ^ { {r \times id }}   X \times X  \rightarrow ^ { m }    & X  \\
{} _  \Delta  \nwarrow  &{}  &\nearrow _  \beta   \\
{}  &X  \mathop \rightarrow \limits _ { { p _ {x} }}  X  &{}  \\
\end{array}

$$

is commutative. Here  $  p _ {X} :  X \rightarrow e $
is the canonical morphism of  $  X $
into the final object  $  e $, 
while  $  \Delta :  X \rightarrow X \times X $
is the diagonal morphism.

If  $  C $
is the category of sets Ens, group objects are precisely groups. The final object of the category Ens is the set  $  \{ e \} $
consisting of the single element  $  e $. 
Axiom a) denotes the associativity of the binary operation given by the morphism  $  m:  X \times X \rightarrow X $. 
The morphism  $  r:  X \rightarrow X $
is the mapping of inversion, while the morphism  $  \beta :  \{ e \} \rightarrow X $
is the mapping of the set  $  \{ e \} $
into  $  X $, 
whose image is equal to the unit element in  $  X $.

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 [[#References|[2]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Demazure,   A. Grothendieck,   "Schémas en groupes I" , ''Lect. notes in math.'' , '''151–153''' , Springer  (1970)</TD></TR></table>

====Comments====
Group objects, in particular categories, are often objects of interest in their own right. For example, topological groups (cf. [[Topological group|Topological group]]) are group objects in the category of topological spaces and continuous mappings; Lie groups (cf. [[Lie group|Lie group]]) are group objects in the category of smooth manifolds; and sheaves of groups on a given space  $  X $
are group objects in the category of sheaves of sets on  $  X $. 
A group object in a category of the form  $   \mathop{\rm Gr} - C $
is an object of  $  C $
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  $   \mathop{\rm Gr} -  \mathop{\rm Gr} - C $
is isomorphic to the category  $   \mathop{\rm Ab} - C $
of Abelian group objects in  $  C $. 
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|fundamental group]] of a topological group is Abelian.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. MacLane,   "Categories for the working mathematician" , Springer  (1971)  pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Eckmann,   P.J. Hilton,   "Group-like structures in general categories I. Multiplications and comultiplications"  ''Math. Ann.'' , '''145'''  (1962)  pp. 227–255</TD></TR></table>