Lurie ∞ operad

Definition 17.4.3. Let \(\Fin _*\) be the category of finite pointed sets. An \(\infty \)-operad in the sense of Lurie is a pair \((\Oo ^{\otimes }, p_{\Oo })\) consisting of an \(\infty \)-category \(\Oo ^{\otimes }\) equipped with a functor \(p_{\Oo }\colon \Oo ^{\otimes } \to \Fin _*\) satisfying the following conditions:

(1)

For any inert morphism \(f\colon I_+ \to J_+\) and any object \(X \in \Oo ^{\otimes }_I\) there is a \(p_{\Oo }\)-cocartesian lift \(X \to Y\) in \(\Oo ^{\otimes }\).

(2)

Let \(X\) and \(Y\) be objects of \(\Oo ^{\otimes }\) and let \(I_+ := p(X)\) and \(J_+ := p(Y)\). For every \(j \in J\), let \(Y \to Y_j\) denote a \(p_{\Oo }\)-cocartesian lift of the Segal map \(\rho _j \colon J_+ \to \{j\}_+\). Then the induced commutative square

is a pullback square.

(3)

For every finite collection \(\{X_j\}_{j \in J}\) of objects \(X_j \in \Oo ^{\otimes }_{\{j\}}\), there exists an object \(X \in \Oo ^{\otimes }_J\) and a collection of \(p_{\Oo }\)-cocartesian morphisms \(X \to X_j\) covering \(\rho _j \colon J_+ \to \{j\}_+\).

We say a functor \(f\colon \Oo ^{\otimes } \to \Pp ^{\otimes }\) over \(\Fin _*\) is a morphism of Lurie-\(\infty \)-operads if it preserves cocartesian morphisms over inerts. We let \(\Op _\infty ^{\mathrm {Lurie}} \subseteq (\Cat _{\infty })_{/\Fin _*}\) denote the \(\infty \)-category of the Lurie \(\infty \)-operads and their morphisms.