Lemma 10.5.3 (Fiberwise one-point compactification). Fiberwise one-point compactification defines a symmetric monoidal functor \[ \Vect _{\R }^{\disc }(X)^{\simeq } \longrightarrow \mathrm {Bun}_{*,\mathrm {c}}(X)^{\simeq }, \qquad E\longmapsto (X\xrightarrow {s_{\infty }}S^E\to X). \]
Proof. The total-space functor from vector bundles over \(X\) to fiber bundles over \(X\) preserves finite products: the total space of \(E\oplus F\) is \(E\times _XF\). Fiberwise one-point compactification carries this product to the fiberwise smash product. More precisely, removal of the section at infinity gives a natural isomorphism \[ (K\wedge _XL)\setminus \{\infty \} \cong (K\setminus \{\infty \})\times _X (L\setminus \{\infty \}). \] Fiberwise one-point compactification and removal of the section at infinity are inverse equivalences between the groupoid of bundles with locally compact Hausdorff fibers and the groupoid \(\mathrm {Bun}_{*,\mathrm {c}}(X)^{\simeq }\). Since removal of the section is fully faithful, the displayed identification is compatible with the associativity, unit, and symmetry constraints. This proves the claim. □
Generated from the authoritative LaTeX source.