zero bundle in nLab

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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↪Bundles 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 BB is the bundle B×{0}→pr 1XB \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 BB is the identity map on BB, 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 XX has all fibers the zero-spectrum 0 •0_\bullet (i.e. the spectrum all whose components are contractible, 0 n≃*0_n \simeq \ast for all nn).

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

• zero object

• bireflective subcategory

• empty bundle