Operadic stabilization

Construction 18.4.1. Let \(\Oo \) be an \(\infty \)-operad that admits finite operadic limits. Consider the \(\infty \)-category \(\An _*^{\fin }\) equipped with the smash product monoidal structure from Definition 16.3.5. We define \[ \oSp (\Oo ) \quad \subseteq \quad \oDay (\Mm _{\An _*^{\fin }}, \Oo ) \] to be the full suboperad whose colors are those functors \(F\colon \An _*^{\fin } \to \Oo _{\lra {1}}\) that are reduced and excisive. Note that \[ \oSp (\Oo )_{\lra {1}} \simeq \Sp ^{\exc }(\Oo _{\lra {1}}) \] by construction; through Proposition 16.5.10, we identify this underlying \(\infty \)-category with the abstract stabilization \(\Sp (\Oo _{\lra {1}})\). The monoidal unit \(S^0\) defines an operad map \(\Comm \to \Mm _{\An _*^{\fin }}\), and restricting the evaluation map along it defines a morphism of \(\infty \)-operads \[ \Omega ^\infty \colon \oSp (\Oo ) \to \Oo . \]