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''in a category''
 
''in a category''
  
A concept in the theory of categories, instances of which are a [[Principal fibre bundle|principal fibre bundle]] in topology, a [[Principal homogeneous space|principal homogeneous space]] in algebraic geometry, etc. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p0747103.png" /> be a [[Group object|group object]] in a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p0747104.png" /> with products and final object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p0747105.png" />. An object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p0747106.png" /> is said to be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p0747107.png" />-object if there is given a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p0747108.png" /> for which the following diagrams are commutative:
+
A concept in the theory of categories, instances of which are a [[Principal fibre bundle|principal fibre bundle]] in topology, a [[Principal homogeneous space|principal homogeneous space]] in algebraic geometry, etc. Let $  G $
 +
be a [[Group object|group object]] in a category $  C $
 +
with products and final object $  e $.  
 +
An object $  P $
 +
is said to be a $  G $-
 +
object if there is given a morphism $  \pi : P \times G \rightarrow P $
 +
for which the following diagrams are commutative:
 +
 
 +
$$
 +
 
 +
\begin{array}{ccc}
 +
P \times G \times G  &\rightarrow ^ { {1 _ P} \times \mu }  &P \times G  \\
 +
size - 3 {\pi \times 1 _ {G} } \downarrow  &{}  &\downarrow size - 3 \pi  \\
 +
P \times G  & \mathop \rightarrow \limits _  \pi    & P  \\
 +
\end{array}
 +
\  \  \  \
 +
 
 +
\begin{array}{ccc}
 +
P \times e  &\rightarrow ^ { {1 _ P} \times \beta }  &P \times G  \\
 +
size - 3 {pr _ {1} } \downarrow  &{}  &\downarrow size - 3 \pi  \\
 +
P  & \mathop \rightarrow \limits _ { {1 _ {P} }}  &P.  \\
 +
\end{array}
  
<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/p/p074/p074710/p0747109.png" /></td> </tr></table>
+
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471010.png" /> is the group law morphism on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471011.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471012.png" /> is the unit element morphism into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471013.png" />. More precisely, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471014.png" />-objects specified as above are called right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471016.png" />-objects; the definition of left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471018.png" />-objects is similar. As an example of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471019.png" />-object one may take the group object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471020.png" /> itself, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471021.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471022.png" />. This object is called the trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471024.png" />-object. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471025.png" />-objects in the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471026.png" /> form a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471027.png" />. The morphisms are morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471028.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471029.png" /> which commute with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471030.png" /> (i.e. such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471031.png" />). A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471032.png" />-object is said to be a formal principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471034.png" />-object if the morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471036.png" /> induce an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471037.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471038.png" /> is some [[Grothendieck topology|Grothendieck topology]] on the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471039.png" />, a formal principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471040.png" />-object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471041.png" /> is called a principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471042.png" />-object (with respect to the topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471043.png" />) if there exists a covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471044.png" /> of the final object such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471045.png" /> the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471046.png" /> is isomorphic to the trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471047.png" />-object.
+
Here $  \mu : G \times G \rightarrow G $
 +
is the group law morphism on $  G $,  
 +
while $  \beta : e \rightarrow G $
 +
is the unit element morphism into $  G $.  
 +
More precisely, the $  G $-
 +
objects specified as above are called right $  G $-
 +
objects; the definition of left $  G $-
 +
objects is similar. As an example of a $  G $-
 +
object one may take the group object $  G $
 +
itself, for which $  \mu $
 +
coincides with $  \pi $.  
 +
This object is called the trivial $  G $-
 +
object. The $  G $-
 +
objects in the category $  C $
 +
form a category $  C  ^ {G} $.  
 +
The morphisms are morphism $  \phi : P \rightarrow P  ^  \prime  $
 +
of $  C $
 +
which commute with $  \pi $(
 +
i.e. such that $  \pi  ^  \prime  ( \phi \times 1 ) = \phi \pi $).  
 +
A $  G $-
 +
object is said to be a formal principal $  G $-
 +
object if the morphisms $  pr _ {1} : P \times G \rightarrow P $
 +
and $  \pi : P \times G \rightarrow P $
 +
induce an isomorphism $  \phi = ( \pi , pr _ {1} ): P \times G \rightarrow P \times P $.  
 +
If $  T $
 +
is some [[Grothendieck topology|Grothendieck topology]] on the category $  C $,  
 +
a formal principal $  G $-
 +
object $  P $
 +
is called a principal $  G $-
 +
object (with respect to the topology $  T $)  
 +
if there exists a covering $  ( U _ {i} \rightarrow e ) _ {i \in I }  $
 +
of the final object such that for any $  i \in I $
 +
the product $  G \times _ {e} U _ {i} $
 +
is isomorphic to the trivial $  G \times _ {e} U _ {i} $-
 +
object.
  
 
===Examples.===
 
===Examples.===
  
 +
1) If  $  C $
 +
is the category of sets and  $  G $
 +
is a group, then the non-empty  $  G $-
 +
objects are called  $  G $-
 +
sets. These are sets  $  P $
 +
for which a mapping  $  P \times G \rightarrow P $(
 +
$  ( p, g) \rightarrow pg $)
 +
is defined such that for any  $  g, g  ^  \prime  \in G $
 +
one has  $  p( g g  ^  \prime  ) = ( pg) g  ^  \prime  $,
 +
and for any  $  p \in P $
 +
it is true that  $  p \cdot 1 = p $.
 +
A principal  $  G $-
 +
object is a  $  G $-
 +
set in which for any  $  p, p  ^  \prime  \in P $
 +
there exists a unique element  $  g \in G $
 +
such that  $  pg = p  ^  \prime  $(
 +
a principal homogeneous  $  G $-
 +
set). If  $  P $
 +
is not empty, the choice of a  $  p _ {0} \in P $
 +
determines a mapping  $  g \rightarrow p _ {0} g $
 +
which establishes an isomorphism between  $  P $
 +
and the trivial  $  G $-
 +
set  $  G $.
 +
Thus, in any topology a formal principal  $  G $-
 +
object is a principal  $  G $-
 +
object.
  
1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471048.png" /> is the category of sets and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471049.png" /> is a group, then the non-empty <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471050.png" />-objects are called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471051.png" />-sets. These are sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471052.png" /> for which a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471053.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471054.png" />) is defined such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471055.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471056.png" />, and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471057.png" /> it is true that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471058.png" />. A principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471059.png" />-object is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471060.png" />-set in which for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471061.png" /> there exists a unique element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471062.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471063.png" /> (a principal homogeneous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471065.png" />-set). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471066.png" /> is not empty, the choice of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471067.png" /> determines a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471068.png" /> which establishes an isomorphism between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471069.png" /> and the trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471070.png" />-set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471071.png" />. Thus, in any topology a formal principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471072.png" />-object is a principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471073.png" />-object.
+
2) If $  X $
 
+
is a differentiable manifold and $  H $
2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471074.png" /> is a differentiable manifold and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471075.png" /> is a Lie group, then, taking <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471076.png" /> to be the category of fibrations over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471077.png" />, taking as group object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471078.png" /> the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471079.png" />, and defining a topology in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471080.png" /> with the aid of families of open coverings, it is possible to obtain the definition of a principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471081.png" />-fibration.
+
is a Lie group, then, taking $  C $
 +
to be the category of fibrations over $  X $,  
 +
taking as group object $  G $
 +
the projection $  H \times X \rightarrow X $,  
 +
and defining a topology in $  C $
 +
with the aid of families of open coverings, it is possible to obtain the definition of a principal $  G $-
 +
fibration.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471082.png" /> is a formal principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471083.png" />-object in a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471084.png" />, then for any object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471085.png" /> in the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471086.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471087.png" /> is either empty or is a principal homogeneous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471088.png" />-set.
+
If $  P $
 +
is a formal principal $  G $-
 +
object in a category $  C $,  
 +
then for any object $  X $
 +
in the category $  \mathop{\rm Ob} ( C) $
 +
the set $  P( X) = \mathop{\rm Hom} _ {C} ( X, P ) $
 +
is either empty or is a principal homogeneous $  G( X) $-
 +
set.
  
A formal principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471089.png" />-object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471090.png" /> is isomorphic to the trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471091.png" />-object if and only if there exists a section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471092.png" />. The set of isomorphism classes of formal principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471093.png" />-objects is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471094.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471095.png" /> is an Abelian group object, then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471096.png" />, with the class of trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471097.png" />-objects as a base point, is a group and can be computed by standard tools of homological algebra. In general, in the computation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471098.png" /> Čech homology constructions are employed (cf. [[Non-Abelian cohomology|Non-Abelian cohomology]]).
+
A formal principal $  G $-
 +
object $  P $
 +
is isomorphic to the trivial $  G $-
 +
object if and only if there exists a section $  e \rightarrow P $.  
 +
The set of isomorphism classes of formal principal $  G $-
 +
objects is denoted by $  H  ^ {1} ( C, G) $.  
 +
If $  G $
 +
is an Abelian group object, then the set $  H  ^ {1} ( C, G ) $,  
 +
with the class of trivial $  G $-
 +
objects as a base point, is a group and can be computed by standard tools of homological algebra. In general, in the computation of $  H  ^ {1} ( C, G) $
 +
Čech homology constructions are employed (cf. [[Non-Abelian cohomology|Non-Abelian cohomology]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Grothendieck (ed.) et al. (ed.) , ''Revêtements étales et groupe fondamental. SGA 1'' , ''Lect. notes in math.'' , '''224''' , Springer (1971) {{MR|0354651}} {{ZBL|1039.14001}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Grothendieck (ed.) et al. (ed.) , ''Revêtements étales et groupe fondamental. SGA 1'' , ''Lect. notes in math.'' , '''224''' , Springer (1971) {{MR|0354651}} {{ZBL|1039.14001}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Formal principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p07471099.png" />-objects are commonly called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p074710101.png" />-torsors. The distinction between formal principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p074710102.png" />-objects and principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p074710103.png" />-objects is not a profound one: a necessary and sufficient condition for a formal principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p074710104.png" />-object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p074710105.png" /> to be principal is that the unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p074710106.png" /> should form a covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074710/p074710107.png" />.
+
Formal principal $  G $-
 +
objects are commonly called $  G $-
 +
torsors. The distinction between formal principal $  G $-
 +
objects and principal $  G $-
 +
objects is not a profound one: a necessary and sufficient condition for a formal principal $  G $-
 +
object $  P $
 +
to be principal is that the unique morphism $  P \rightarrow e $
 +
should form a covering of $  e $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Giraud, "Cohomologie non abélienne" , Springer (1971) {{MR|0344253}} {{ZBL|0226.14011}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Giraud, "Cohomologie non abélienne" , Springer (1971) {{MR|0344253}} {{ZBL|0226.14011}} </TD></TR></table>

Revision as of 14:54, 7 June 2020


in a category

A concept in the theory of categories, instances of which are a principal fibre bundle in topology, a principal homogeneous space in algebraic geometry, etc. Let $ G $ be a group object in a category $ C $ with products and final object $ e $. An object $ P $ is said to be a $ G $- object if there is given a morphism $ \pi : P \times G \rightarrow P $ for which the following diagrams are commutative:

$$ \begin{array}{ccc} P \times G \times G &\rightarrow ^ { {1 _ P} \times \mu } &P \times G \\ size - 3 {\pi \times 1 _ {G} } \downarrow &{} &\downarrow size - 3 \pi \\ P \times G & \mathop \rightarrow \limits _ \pi & P \\ \end{array} \ \ \ \ \begin{array}{ccc} P \times e &\rightarrow ^ { {1 _ P} \times \beta } &P \times G \\ size - 3 {pr _ {1} } \downarrow &{} &\downarrow size - 3 \pi \\ P & \mathop \rightarrow \limits _ { {1 _ {P} }} &P. \\ \end{array} $$

Here $ \mu : G \times G \rightarrow G $ is the group law morphism on $ G $, while $ \beta : e \rightarrow G $ is the unit element morphism into $ G $. More precisely, the $ G $- objects specified as above are called right $ G $- objects; the definition of left $ G $- objects is similar. As an example of a $ G $- object one may take the group object $ G $ itself, for which $ \mu $ coincides with $ \pi $. This object is called the trivial $ G $- object. The $ G $- objects in the category $ C $ form a category $ C ^ {G} $. The morphisms are morphism $ \phi : P \rightarrow P ^ \prime $ of $ C $ which commute with $ \pi $( i.e. such that $ \pi ^ \prime ( \phi \times 1 ) = \phi \pi $). A $ G $- object is said to be a formal principal $ G $- object if the morphisms $ pr _ {1} : P \times G \rightarrow P $ and $ \pi : P \times G \rightarrow P $ induce an isomorphism $ \phi = ( \pi , pr _ {1} ): P \times G \rightarrow P \times P $. If $ T $ is some Grothendieck topology on the category $ C $, a formal principal $ G $- object $ P $ is called a principal $ G $- object (with respect to the topology $ T $) if there exists a covering $ ( U _ {i} \rightarrow e ) _ {i \in I } $ of the final object such that for any $ i \in I $ the product $ G \times _ {e} U _ {i} $ is isomorphic to the trivial $ G \times _ {e} U _ {i} $- object.

Examples.

1) If $ C $ is the category of sets and $ G $ is a group, then the non-empty $ G $- objects are called $ G $- sets. These are sets $ P $ for which a mapping $ P \times G \rightarrow P $( $ ( p, g) \rightarrow pg $) is defined such that for any $ g, g ^ \prime \in G $ one has $ p( g g ^ \prime ) = ( pg) g ^ \prime $, and for any $ p \in P $ it is true that $ p \cdot 1 = p $. A principal $ G $- object is a $ G $- set in which for any $ p, p ^ \prime \in P $ there exists a unique element $ g \in G $ such that $ pg = p ^ \prime $( a principal homogeneous $ G $- set). If $ P $ is not empty, the choice of a $ p _ {0} \in P $ determines a mapping $ g \rightarrow p _ {0} g $ which establishes an isomorphism between $ P $ and the trivial $ G $- set $ G $. Thus, in any topology a formal principal $ G $- object is a principal $ G $- object.

2) If $ X $ is a differentiable manifold and $ H $ is a Lie group, then, taking $ C $ to be the category of fibrations over $ X $, taking as group object $ G $ the projection $ H \times X \rightarrow X $, and defining a topology in $ C $ with the aid of families of open coverings, it is possible to obtain the definition of a principal $ G $- fibration.

If $ P $ is a formal principal $ G $- object in a category $ C $, then for any object $ X $ in the category $ \mathop{\rm Ob} ( C) $ the set $ P( X) = \mathop{\rm Hom} _ {C} ( X, P ) $ is either empty or is a principal homogeneous $ G( X) $- set.

A formal principal $ G $- object $ P $ is isomorphic to the trivial $ G $- object if and only if there exists a section $ e \rightarrow P $. The set of isomorphism classes of formal principal $ G $- objects is denoted by $ H ^ {1} ( C, G) $. If $ G $ is an Abelian group object, then the set $ H ^ {1} ( C, G ) $, with the class of trivial $ G $- objects as a base point, is a group and can be computed by standard tools of homological algebra. In general, in the computation of $ H ^ {1} ( C, G) $ Čech homology constructions are employed (cf. Non-Abelian cohomology).

References

[1] A. Grothendieck (ed.) et al. (ed.) , Revêtements étales et groupe fondamental. SGA 1 , Lect. notes in math. , 224 , Springer (1971) MR0354651 Zbl 1039.14001

Comments

Formal principal $ G $- objects are commonly called $ G $- torsors. The distinction between formal principal $ G $- objects and principal $ G $- objects is not a profound one: a necessary and sufficient condition for a formal principal $ G $- object $ P $ to be principal is that the unique morphism $ P \rightarrow e $ should form a covering of $ e $.

References

[a1] J. Giraud, "Cohomologie non abélienne" , Springer (1971) MR0344253 Zbl 0226.14011
How to Cite This Entry:
Principal G-object. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Principal_G-object&oldid=49375
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article