empty bundle in nLab

Contents

Contents

Idea

In as far as a general bundle over some base object BB is a morphism E→BE \to B into BB, out of some “total space” object EE, the empty bundle over BB is the one whose total space is “empty”, hence whose projection map is the empty function.

Definition

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.:

∅ ↓ B. \array{ \varnothing \\ \big\downarrow \\ B \mathrlap{\,.} }

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

(1)∃(X→∅)⇔X≃∅. \exists \big( X \overset{}{\rightarrow} \varnothing \big) \;\;\;\;\; \Leftrightarrow \;\;\;\;\; X \simeq \varnothing \,.

Properties

In topological spaces

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:

  B×∅≃∅ B \times \varnothing \;\simeq\; \varnothing

• 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):

  D n ⟶¬∃ ∅ ↓ ↓ D n×I ⟶ B \array{ D^{n} &\overset{ \not \exists }{\longrightarrow}& \varnothing \\ \big\downarrow && \big\downarrow \\ D^n \times I &\longrightarrow& B }

In simplicial sets

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):

  Λ k n ⟶¬∃ ∅ ↓ ↓ Δ n ⟶ B \array{ \Lambda_k^n &\overset{ \not \exists }{\longrightarrow}& \varnothing \\ \big\downarrow && \big\downarrow \\ \Delta^n &\longrightarrow& B }

  (keeping in mind that the 0-simplex has no horns, hence that all horns are inhabited).

Application

In equivariant bundle theory

Empty fiber bundles play a central role in the context of equivariant bundles, where they frequently appear as fixed loci of non-empty bundles.

• zero bundle