Tensor product vector bundles symmetric monoidal

Proof. Let \(\alpha \colon S_+ \to T_+\) be a pointed map and let \((V_s)_{s \in S}\) be an object over \(S_+\). For each \(t \in T\), choose the tensor product \[ W_t := \bigotimes _{s \in \alpha ^{-1}(t)} V_s, \] with the convention that the tensor product over the empty set is the trivial line bundle \(\ul {\C }\). Let \[ \mu _t\colon \prod _{s \in \alpha ^{-1}(t)} V_s \to W_t \] be the universal multilinear map. Local trivializations show that \(W_t\) is again a vector bundle and that \(\mu _t\) is continuous. These maps define a morphism \[ (V_s)_{s \in S} \longrightarrow (W_t)_{t \in T} \] over \(\alpha \).

We claim that this morphism is \(p_X\)-cocartesian. Let \(\beta \colon T_+ \to U_+\) be another pointed map, and let \((Z_u)_{u \in U}\) be an object over \(U_+\). By definition, a morphism from \((W_t)_{t \in T}\) to \((Z_u)_{u \in U}\) over \(\beta \) is a family of multilinear maps \[ \prod _{t \in \beta ^{-1}(u)} W_t \to Z_u. \] Precomposition with the maps \(\mu _t\) identifies this set with the set of multilinear maps \[ \prod _{s \in (\beta \alpha )^{-1}(u)} V_s \to Z_u, \] for all \(u \in U\), by the universal property of the tensor products \(W_t\). This is precisely the required universal property of a cocartesian morphism.

The fiber over \(S_+\) is visibly \(\Vect ^{\disc }(X)^S\), so the Segal maps are isomorphisms. Thus \(p_X\) is a symmetric monoidal category in Lurie’s model. Taking nerves gives the corresponding cocartesian fibration of \(\infty \)-categories, and the equivalence of operad models from Proposition 17.4.8 carries it to the desired symmetric monoidal \(\infty \)-category over \(\Span (\Fin )\). Its tensor product is obtained by pushing forward along the active map \(\{1,2\}_+ \to \{1\}_+\), hence is the usual tensor product of vector bundles. □