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''of a category''
 
''of a category''
  
An object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g0452501.png" /> of a [[Category|category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g0452502.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g0452503.png" /> the set of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g0452504.png" /> is a group, while the correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g0452505.png" /> is a functor from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g0452506.png" /> into the category of groups Gr. A homomorphism of a group object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g0452507.png" /> into a group object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g0452508.png" /> is a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g0452509.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525010.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525011.png" /> the corresponding mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525012.png" /> is a homomorphism of groups. The group objects of a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525013.png" /> and homomorphisms between them form the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525014.png" />. The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525015.png" /> establishes an equivalence between the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525016.png" /> and the category of representable pre-sheaves of groups on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525017.png" />. If the values of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525018.png" /> belong to the subcategory Ab of Abelian groups, then the group object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525019.png" /> is said to be commutative or Abelian. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525020.png" /> has finite products and a final object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525021.png" />, a group object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525022.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525023.png" /> is defined by the following properties.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525024.png" /> (multiplication), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525025.png" /> (inversion) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525026.png" /> (a unit) satisfying the following axioms.
+
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
 
Associativity. The diagram
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525027.png" /></td> </tr></table>
+
$$
 +
 
 +
\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.
 
is commutative.
Line 13: Line 59:
 
Existence of a unit element. The diagram
 
Existence of a unit element. The diagram
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525028.png" /></td> </tr></table>
+
$$
 +
 
 +
\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.
 
is commutative.
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Existence of an inverse element. The diagram
 
Existence of an inverse element. The diagram
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525029.png" /></td> </tr></table>
+
$$
  
is commutative. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525030.png" /> is the canonical morphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525031.png" /> into the final object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525032.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525033.png" /> is the diagonal morphism.
+
\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}
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525034.png" /> is the category of sets Ens, group objects are precisely groups. The final object of the category Ens is the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525035.png" /> consisting of the single element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525036.png" />. Axiom a) denotes the associativity of the binary operation given by the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525037.png" />. The morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525038.png" /> is the mapping of inversion, while the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525039.png" /> is the mapping of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525040.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525041.png" />, whose image is equal to the unit element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525042.png" />.
+
$$
 +
 
 +
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]]].
 
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]]].
Line 29: Line 105:
 
====References====
 
====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>
 
<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====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525043.png" /> are group objects in the category of sheaves of sets on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525044.png" />. A group object in a category of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525045.png" /> is an object of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525046.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525047.png" /> is isomorphic to the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525048.png" /> of Abelian group objects in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045250/g04525049.png" />. 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.
+
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====
 
====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>
 
<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>

Latest revision as of 19:42, 5 June 2020


of a category

An object $ X $ of a 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 [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 $ 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 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
How to Cite This Entry:
Group object. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Group_object&oldid=47144
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article