Difference between revisions of "Principal G-object"

Revision as of 14:52, 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 be a group object in a category with products and final object . An object is said to be a -object if there is given a morphism for which the following diagrams are commutative:

Here is the group law morphism on , while is the unit element morphism into . More precisely, the -objects specified as above are called right -objects; the definition of left -objects is similar. As an example of a -object one may take the group object itself, for which coincides with . This object is called the trivial -object. The -objects in the category form a category . The morphisms are morphism of which commute with (i.e. such that ). A -object is said to be a formal principal -object if the morphisms and induce an isomorphism . If is some Grothendieck topology on the category , a formal principal -object is called a principal -object (with respect to the topology ) if there exists a covering of the final object such that for any the product is isomorphic to the trivial -object.

Examples.

1) If is the category of sets and is a group, then the non-empty -objects are called -sets. These are sets for which a mapping () is defined such that for any one has , and for any it is true that . A principal -object is a -set in which for any there exists a unique element such that (a principal homogeneous -set). If is not empty, the choice of a determines a mapping which establishes an isomorphism between and the trivial -set . Thus, in any topology a formal principal -object is a principal -object.

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

If is a formal principal -object in a category , then for any object in the category the set is either empty or is a principal homogeneous -set.

A formal principal -object is isomorphic to the trivial -object if and only if there exists a section . The set of isomorphism classes of formal principal -objects is denoted by . If is an Abelian group object, then the set , with the class of trivial -objects as a base point, is a group and can be computed by standard tools of homological algebra. In general, in the computation of 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 -objects are commonly called -torsors. The distinction between formal principal -objects and principal -objects is not a profound one: a necessary and sufficient condition for a formal principal -object to be principal is that the unique morphism should form a covering of .

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=48287

This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article

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 ''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  $  G $
+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:
-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:
-$$
+<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  $  \mu :  G \times G \rightarrow G $
+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.
-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.===
-) 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.
-) If  $  X $
+) 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.
-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 $
+) 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 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 $-
+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.
-object  $  P $
-is isomorphic to the trivial  $  G $-
+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]]).
-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====
 <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====
-Formal principal  $  G $-
+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" />.
-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====
 <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>

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Difference between revisions of "Principal G-object"

Revision as of 14:52, 7 June 2020

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