Stabilization of operads

Proof. The first claim is Proposition 18.4.3. We now prove the adjunction. Postcomposition with a map in \(\Op _{\infty }^{\lex }\) preserves reduced excisive functors, since its color functor preserves finite limits. Thus the construction of \(\oSp (\Oo )\) is functorial in \(\Oo \in \Op _{\infty }^{\lex }\), and evaluation at \(S^0\) gives a natural transformation \[ \Omega ^\infty \colon \oSp \to \id _{\Op _{\infty }^{\lex }}. \] By the dual of the recognition criterion for Bousfield localizations from Proposition 21.8.9, applied in \((\Op _{\infty }^{\lex })\catop \), it is enough to show that \(\Omega ^\infty _{\Oo }\) is an equivalence whenever \(\Oo \) is stable, and that \[ \oSp (\Omega ^\infty _{\Oo })\colon \oSp (\oSp (\Oo )) \to \oSp (\Oo ) \] is an equivalence for every \(\Oo \in \Op _{\infty }^{\lex }\). The first assertion is Proposition 18.4.5.

For the second assertion, observe that \(\oSp (\oSp (\Oo ))\) may be identified with the full suboperad of \(\oDay (\Mm _{\An _*^{\fin } \times \An _*^{\fin }},\Oo )\) spanned by bifunctors \(\An _*^{\fin } \times \An _*^{\fin } \to \Oo _{\lra {1}}\) which are reduced and excisive in each variable. Under this identification, the map \(\oSp (\Omega ^\infty _{\Oo })\) evaluates in one variable at \(S^0\), while the map \[ \Omega ^\infty _{\oSp (\Oo )}\colon \oSp (\oSp (\Oo )) \to \oSp (\Oo ) \] evaluates in the other variable at \(S^0\). These two maps are conjugate by the symmetry of \(\An _*^{\fin }\times \An _*^{\fin }\). Since \(\oSp (\Oo )\) is stable by the first part of the theorem, Proposition 18.4.5 shows that \(\Omega ^\infty _{\oSp (\Oo )}\) is an equivalence. Hence \(\oSp (\Omega ^\infty _{\Oo })\) is an equivalence as well. □