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) 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