Classes of bundles
Examples and Applications
Special and general types
Let be an ambient (∞,1)-topos. Let be two objects of . Then a -fiber bundle over in is a morphism such that there is an effective epimorphism and an (∞,1)-pullback square of the form
Externally this is a -fiber -bundle.
See at associated ∞-bundle for more.
A fiber -bundle whose typical fiber is a pointed connected object, hence a delooping of an ∞-group
is a -∞-gerbe.
Every -fiber -bundle is the associated ∞-bundle to an automorphism ∞-group-principal ∞-bundle.
For let be the object classifier. Then any bundle is classified by a morphism
On the other hand, since the pullback to the bundle on some is trivializable, that bundle over is classsified by a map that factors through the point which is the name of the fiber
The 1-image-of this point inclusion is the delooping of the automorphism ∞-group of :
Therefore the fact that is trivialized over means that there the classifying maps fit into a commuting diagram of the form
By assumption the left morphism is a 1-epimorphism and by the above construction the right morphism is a 1-monomorphism. Therefore by the (n-connected, n-truncated) factorization system this diagram has an essentially unique lift
This diagonal lift classifies an -principal ∞-bundle and the commutativity of the bottom right triangle exhibits the original bundle as the associated ∞-bundle to that.
See the references at associated ∞-bundle.
The explicit general definition appears as def. 4.1 in part I of
Revised on November 25, 2014 19:39:29
by Urs Schreiber