Heuristic definition of an ∞-operad

Definition 12.4.3. An \(\infty \)-operad is a pair \(\Oo = (\Oo ^{\otimes },p_{\Oo })\) consisting of an \(\infty \)-category \(\Oo ^{\otimes }\) equipped with a functor \(p_{\Oo }\colon \Oo ^{\otimes } \to \Span (\Fin )\) satisfying the following conditions:

(1)

The \(\infty \)-category \(\Oo ^{\otimes }\) admits finite products, and the functor \(p_{\Oo }\) preserves finite products;

(2)

For every finite set \(I\), the product in \(\Oo ^{\otimes }\) defines an equivalence \[ \prod _{i \in I} \Oo ^{\otimes }_{\{i\}} \iso \Oo ^{\otimes }_I. \] Here we write \(\Oo ^{\otimes }_I\) for the fiber of \(p_{\Oo }\) over the set \(I\).

(3)

For a morphism \(f\colon J \to I\) in \(\Fin \) and objects \(X_i \in \Oo ^{\otimes }\), the map \(\widetilde {f}\colon \prod _{i\in I} X_i \to \prod _{j \in J} X_{f(j)}\) whose \(j\)-th component is the projection to \(X_{f(j)}\) is \(p_{\Oo }\)-cocartesian, in the sense that for every \(Y \in \Oo ^{\otimes }\) the following square is a pullback square: