Lemma 10.5.4 (Parametrized realization). The functor \(\Pi _{\infty ,X}\) preserves finite products. On pointed compact bundles it induces a canonical lax symmetric monoidal functor \[ \Pi _{\infty ,X,*}\colon \big (\mathrm {Bun}_{*,\mathrm {c}}(X),\wedge _X\big ) \longrightarrow \big ((\An _{/\Pi _{\infty }(X)})_*,\wedge _{\Pi _{\infty }(X)}\big ). \]
Proof. The terminal bundle is sent to the terminal object \(\Pi _{\infty }(X)\to \Pi _{\infty }(X)\). If \(E\to X\) and \(F\to X\) are fiber bundles, their product is \(E\times _XF\to X\). Since every fiber bundle is a Serre fibration by Theorem 2.3.13, we obtain from Proposition 2.3.19 a natural isomorphism \[ \Pi _{\infty }(E\times _XF) \cong \Pi _{\infty }(E) \times _{\Pi _{\infty }(X)} \Pi _{\infty }(F). \] Thus \(\Pi _{\infty ,X}\) preserves finite products. Straightening identifies \(\An _{/\Pi _{\infty }(X)}\) with \(\Fun (\Pi _{\infty }(X),\An )\), so it has finite colimits and its cartesian product preserves them separately in both variables. The lax symmetric monoidal refinement on pointed objects is therefore supplied by Lemma 10.5.1. On compact pointed bundles, the tensor operations in this pointed construction are represented by fiberwise smash products: a bundle map \(\prod _{i=1}^rK_i\to L\) which sends the union on which some coordinate is the section at infinity to the section at infinity factors uniquely through a pointed bundle map \[ K_1\wedge _X\dots \wedge _XK_r\longrightarrow L. \] Restricting pointed realization to these bundles therefore gives the asserted lax symmetric monoidal functor. □
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