(see also Chern-Weil theory, parameterized homotopy theory)
In as far as a general bundle over some base object $B$ is a morphism $E \to B$ into $B$, out of some “total space” object $E$, the empty bundle over $B$ is the one whose total space is “empty”, hence whose projection map is the empty function.
In TopologicalSpaces an empty bundle has the empty topological space as its total space, while in SimplicialSets the empty bundle has the the empty simplicial set as its total space, etc.:
Generally, one may speak of empty bundles internal to any ambient category in which the initial object is strict (e.g. any topos) in that every morphism to the initial object is an isomorphism, so that
In TopologicalSpaces, any empty bundle
is a locally trivial fiber bundle with typical fiber the empty topological space,
because any product topological space with the empty topological space is itself empty:
is a Serre fibration (in fact a Hurewicz fibration),
because none of the commuting squares that one would have to lift in actually exist, by (1):
Similarly, in SimplicialSets every empty bundle
is a Kan fibration,
since none of the commuting squares that one would have to lift in actually exist, by (1):
(keeping in mind that the 0-simplex has no horns, hence that all horns are inhabited).
Empty fiber bundles play a central role in the context of equivariant bundles, where they frequently appear as fixed loci of non-empty bundles.
Created on April 16, 2021 at 01:21:42. See the history of this page for a list of all contributions to it.