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 $ 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