Principal G-object
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) |
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) |
Principal G-object. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Principal_G-object&oldid=14516