Symmetric monoidal category vector bundles

Construction 19.7.5. For a finite pointed set \(S_+\), we write \(S\) for the complement of the basepoint. Define a 1-category \[ \Vect ^{\otimes ,\disc }(X) \] equipped with a functor \(p_X\colon \Vect ^{\otimes ,\disc }(X) \to \Fin _*\) as follows. An object over \(S_+\) is an \(S\)-indexed family \((V_s)_{s \in S}\) of vector bundles over \(X\). A morphism over a pointed map \(\alpha \colon S_+ \to T_+\) from \((V_s)_{s \in S}\) to \((W_t)_{t \in T}\) consists of, for every \(t \in T\), a map over \(X\) \[ \phi _t\colon \prod _{s \in \alpha ^{-1}(t)} V_s \longrightarrow W_t \] which is continuous and complex multilinear on each fiber. Here the product is the fiber product over \(X\), and if \(\alpha ^{-1}(t)=\emptyset \) we interpret the source as \(X\), so that \(\phi _t\) is the same as a section of \(W_t\).

Composition is substitution of multilinear maps. Thus, if \(\beta \colon T_+ \to U_+\) is another pointed map and \(\psi _u\colon \prod _{t \in \beta ^{-1}(u)}W_t \to Z_u\) is the corresponding family of multilinear maps, the composite over \(\beta \alpha \) is given by \[ (v_s)_{s \in (\beta \alpha )^{-1}(u)} \longmapsto \psi _u\left ( \left (\phi _t((v_s)_{s \in \alpha ^{-1}(t)})\right )_{t \in \beta ^{-1}(u)} \right ). \] This is again multilinear in all variables. The identity morphism over \(\id _{S_+}\) is given by the identity maps \(V_s \to V_s\).