nLab external direct sum of vector bundles

Idea

The external direct sum or external Whitney sum of vector bundles is the operation which from a pair of vector bundles $\mathscr{V} \,\in\,$ Vect(X), $\mathscr{W} \,\in\,$ $Vect(Y)$ over possibly different base spaces $X,\ Y \,\in\,$ Top forms the direct sum of vector bundles (“Whitney sum”) $(-)\bigoplus(-)$ of their pullback? to the product space $X \times Y$ of their base spaces:

$$\array{ Vect(X) \times Vect(Y) &\oversey{\boxplus}{\longrightarrow}& Vect(X\times Y) \\ \big( \mathscr{V} ,\, \mathscr{W} \big) &\mapsto& \big( (pr_X)^\ast \mathscr{V} \big) \oplus_{X \times Y} \big( (pr_Y)^\ast \mathscr{W} \big) }$$

Since the Whitney sum $\oplus \,\colon\, Vect(X \times Y) \times Vect(X \times Y) \longrightarrow Vect(X \times Y)$ is the Cartesian product in $Vect(X \times Y)$, one may understand the external Whitney sum as the Cartesian product in the Grothendieck construction $\int_{X} Vect(X)$, see the the example there.

Related concepts

• external tensor product

• external tensor product of vector bundles

• direct sum of vector bundles

References

Discussion in the context of topological K-theory:

• Max Karoubi, §4.9 in: K-Theory – An introduction, Grundlehren der mathematischen Wissenschaften 226 Springer (1978) [pdf, doi:10.1007%2F978-3-540-79890-3]