nLab zero bundle

Idea

For any notion of bundles with fibers from a pointed category, a zero bundle is a bundle all whose fibers are zero objects.

The construction of zero-bundles typically (such as in the following examples) constitutes a bireflective subcategory inclusion

$$Spaces \hookrightarrow Bundles$$

of the category of base spaces into the given category of bundles.

For example:

• in the context of vector bundles the zero-bundle over a base space $B$ is the bundle $B \times \{0\} \xrightarrow{pr_1} X$ all whose fibers are the zero-dimensional vector space, and the induced bireflective subcategory inclusion is that of base spaces into the corresponding category VectBund;

• in the context of retractive spaces the zero-bundle over a base space $B$ is the identity map on $B$, all whose fibers are (equal or, in homotopy theory, equivalent to) the point $\ast$ regarded as a pointed space;

• in the context of parameterized spectra, the zero-bundle over a base space $X$ has all fibers the zero-spectrum $0_\bullet$ (i.e. the spectrum all whose components are contractible, $0_n \simeq \ast$ for all $n$).

(Beware that a zero-bundle is generally – such as in the above examples – not an empty bundle.)

Related concepts

• zero object

• bireflective subcategory

• empty bundle