# 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:

$$\begin{array}{ccc} P \times G \times G &\rightarrow ^ { {1 _ P} \times \mu } &P \times G \\ {\pi \times 1 _ {G} } \downarrow &{} &\downarrow \pi \\ P \times G & \mathop \rightarrow \limits _ \pi & P \\ \end{array} \ \ \ \ \begin{array}{ccc} P \times e &\rightarrow ^ { {1 _ P} \times \beta } &P \times G \\ {pr _ {1} } \downarrow &{} &\downarrow \pi \\ P & \mathop \rightarrow \limits _ { {1 _ {P} }} &P. \\ \end{array}$$

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.

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### 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).

How to Cite This Entry:
Principal G-object. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Principal_G-object&oldid=49654
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article