Lurie operads are F operads

Proof. Condition (1) in the two definitions is the same. Suppose first that \(p_{\Oo }\colon \Oo ^\otimes \to \Fin _*\) is an \(\infty \)-operad in the sense of Lurie. The functor \[ \Oo ^\otimes _J \longrightarrow \prod _{j \in J}\Oo ^\otimes _{\{j\}} \] induced by cocartesian transport along the Segal maps is essentially surjective by condition (3) of Definition 17.4.3. It is fully faithful as well: condition (2), restricted to the fiber over \(\id _{J_+}\), identifies the hom anima in \(\Oo ^\otimes _J\) with the product of the hom animae in the fibers over the singletons, using cocartesian transport along the Segal maps on the sources. Thus condition (2) holds for the decomposition of \(J\) into singletons, and hence for every finite decomposition by grouping the singleton factors. Condition (3) follows similarly from condition (2): apply it to the singleton decomposition of \(J\) and to those of the summands \(J_i\), and group the resulting factors according to \(J = \bigsqcup _i J_i\).

Conversely, if \(\Oo \) is an \(\Ff _{\mathrm {Lurie}}\)-operad, condition (1) gives the required cocartesian lifts of inert maps. Specializing condition (3) to the decomposition of \(J\) into singletons gives condition (2) of Definition 17.4.3, while condition (2) gives condition (3). Finally, both notions of morphism require precisely the preservation of cocartesian morphisms over inert maps. □