Suspension spectrum functor symmetric monoidal

Proof. We will show that \(\Omega ^{\infty }\colon \Sp \to \An \) is lax symmetric monoidal and that it has a strong monoidal left adjoint.

Under the equivalence \(\Sp \simeq \Exc _*(\An ^{\fin }_*,\An )\), we may identify \(\Omega ^{\infty }\) with the composite \[ \Sp \hookrightarrow \Fun (\An ^{\fin }_*,\An ) \xrightarrow {\ev _{S^0}} \An . \] The first functor is lax symmetric monoidal by construction of the tensor product of \(\Sp \). The second is lax symmetric monoidal by the precomposition statement of Proposition 16.2.10, applied to the symmetric monoidal functor \(S^0\colon *\to \An _*^{\fin }\).

The left adjoint of the first functor is symmetric monoidal by construction of the tensor product on \(\Sp \). Since \(\An _*^{\fin }\) is small and the cartesian product of animae preserves small colimits separately in both variables, Lemma 16.2.11 shows that left Kan extension along \(S^0\colon *\to \An _*^{\fin }\) is symmetric monoidal. The left adjoint \(\S [-]\) of the displayed composite is therefore symmetric monoidal. □