Corollary 19.7.7. For every topological space \(X\), the groupoid \[ \Vect ^{\disc }(X)^{\simeq } \] of finite-rank complex vector bundles and bundle isomorphisms is naturally an object of \(\CRig (\An )\). This structure is contravariantly functorial in \(X\) by pullback.
Proof. Direct sum gives finite coproducts in the small 1-category \(\Vect ^{\disc }(X)\), with initial object the rank-zero bundle \(\ul {0}\). Tensor product distributes over direct sums in each variable, so Corollary 19.7.4 applies to the symmetric monoidal structure of Proposition 19.7.6. This construction is natural in symmetric monoidal functors which preserve finite coproducts. Since pullback preserves direct sums, tensor products, and trivial bundles, it gives the asserted functoriality in \(X\). □
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