Difference between revisions of "Principal G-object"
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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 | + | 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 | ||
− | 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: | ||
− | + | <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 | + | 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 | ||
− | 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|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.=== | ===Examples.=== | ||
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− | + | 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. | |
− | is | ||
− | is a | ||
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− | If | + | 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 | ||
− | |||
− | then | ||
− | |||
− | the | ||
− | |||
− | |||
− | A formal principal | + | 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 | + | |
− | is isomorphic to the trivial | + | 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 | ||
− | 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|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 | + | 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 | ||
− | 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==== | ====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 14:52, 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 |
Principal G-object. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Principal_G-object&oldid=49375