Given a right principal -bundle and a left -action on some , all in a sufficiently strong category (such as Top), one can form the quotient object , where is a product and is the smallest congruence such that (using generalized elements) ; there is a canonical projection where the class of is mapped to , hence making into a fibre bundle with typical fiber , and the transition functions belonging to the action of on . We say that is the associated bundle to with fiber .
In the context of higher topos theory there is an elegant and powerful definition and construction of associated bundles. We discuss here some basics and how this recovers the traditional definition. For more see at associated infinity-bundle and at geometry of physics -- representations and associated bundles.
At geometry of physics -- principal bundles in the section Smooth principal bundles via smooth groupoids is discussed how smooth principal bundles for a Lie group over a smooth manifold are equivalently the homotopy fibers of morphisms of smooth groupoids (smooth stacks) of the form
and so it is natural to construct their homotopy fiber product.
We now discuss that this is equivalently the associated bundle which is associated to the principal bundle via the action .
For a smooth group (e.g. a Lie group), a smooth manifold, a smooth -principal bundle over and a smooth action of on some smooth manifold , then the associated -fiber bundle is equivalently (regarded as a smooth groupoid) the homotopy pullback of the action groupoid-projection along a morphism which modulates
By the discussion at geometry of physics -- principal bundles in the section Smooth principal bundles via smooth groupoids, the morphism of smooth groupoids is presented by a morphism of pre-smooth groupoids after choosing an open cover over wich trivialize and choosing a trivialization, by the zig-zag
Moreover, by this proposition the morphism is a global fibration of pre-smooth groupoids, hence, by the discussion at geometry of physics -- smooth homotopy types, the homotopy pullback in question is equivalently computed as the ordinary pullback of pre-smooth groupoids of along this
This pullback is computed componentwise. Hence it is the pre-smooth groupoid whose morphisms are pairs consisting of a morphism in the Cech groupoid as well as a morphism in the action groupoid, such that the group label of the latter equals the cocycle of the cocycle on the former. Schematically:
This means that the pullback groupoid has at most one morphism between every ordered pair of objects. Accordingly this groupoid is equivalence of groupoids equivalent to the quotient of its space of objects by the equivalence relation induced by its morphisms:
This is a traditional description of the associated bundle in question.