Full suboperad

Construction 14.5.1. Let \(\Oo \) be an \(\infty \)-operad. Given a subanima \(\Pp ^{\simeq } \subseteq \Oo ^{\simeq }\), let \(\Pp ^{\otimes } \subseteq \Oo ^{\otimes }\) be the full subcategory spanned by objects \(\{x_i\}_{i \in I}\) with \(x_i \in \Pp ^{\simeq }\) for all \(i\). Observe that the composite \(p_{\Pp }\colon \Pp ^{\otimes } \hookrightarrow \Oo ^{\otimes } \xrightarrow {p_{\Oo }} \Span (\Fin )\) exhibits \(\Pp ^{\otimes }\) as an \(\infty \)-operad and the inclusion \(\Pp ^{\otimes } \hookrightarrow \Oo ^{\otimes }\) is a morphism of \(\infty \)-operads:

(1)

\(\Pp ^{\otimes }\) admits products inherited from \(\Oo ^{\otimes }\);

(2)

By definition the equivalence \(\prod _{i \in I} \Oo ^{\otimes }_{\{i\}} \iso \Oo ^{\otimes }_I\) restricts to \(\prod _{i \in I} \Pp ^{\otimes }_{\{i\}} \iso \Pp ^{\otimes }_I\);

(3)

By full faithfulness, \(p_{\Oo }\)-cocartesian morphisms \(\widetilde {f}\colon \prod _{i \in I} X_i \to \prod _{j \in J} X_{f(j)}\) in \(\Oo ^{\otimes }\) are still \(p_{\Pp }\)-cocartesian as morphisms in \(\Pp ^{\otimes }\).

We refer to the pair \(\Pp = (\Pp ^{\otimes }, p_{\Pp })\) as the full suboperad of \(\Oo \) spanned by the colors \(\Pp ^{\simeq }\).