\Ff-operad

Definition 17.3.10 (\(\Ff \)-operad). Let \(\Ff = (F,F_L,F_R)\) be an adequate triple satisfying the conditions of Convention 17.3.1. An \(\Ff \)-operad is a pair \(\Oo = (\Oo ^{\otimes },p_{\Oo })\) consisting of an \(\infty \)-category \(\Oo ^{\otimes }\) and a functor \(p_{\Oo }\colon \Oo ^{\otimes } \to \Span _{L,R}(F)\) satisfying the following three conditions:

(1)

The functor \(p_{\Oo }\) is an \(\Ll \)-cocartesian fibration;

(2)

The straightening \(F_L\catop \to \Cat _{\infty }\) of the resulting cocartesian fibration \(\Oo ^{\otimes }\vert _{F_L\catop } \to F_L\catop \) satisfies the Segal condition: for all \(J_1, \dots , J_n \in F\) we have \[ \Oo ^{\otimes }_{\bigsqcup _{i=1}^n J_i} \iso \prod _{i=1}^n \Oo ^{\otimes }_{J_i}. \]

(3)

Consider objects \(I, J_1, \dots , J_n \in F\), \(X \in \Oo ^{\otimes }_I\) and \(Y \in \Oo ^{\otimes }_J\) with \(J := \bigsqcup _{i=1}^n J_i\). Let \(Y \to Y_i\) be cocartesian lifts of the backwards maps \(J \hookleftarrow J_i \xrightarrow {=} J_i\). Then the induced commutative square

is a pullback square.

Given another \(\Ff \)-operad \(\Pp = (\Pp ^{\otimes },p_{\Pp })\), a morphism of \(\Ff \)-operads is a \(\Ll \)-cocartesian functor \(f\colon \Oo ^{\otimes } \to \Pp ^{\otimes }\) over \(\Span _{L,R}(F)\). We denote by \[ \Op _{\Ff } \subseteq (\Cat _\infty )^{\Ll \mathrm {-cocart}}_{/\Span _{L,R}(F)} \] the full subcategory spanned by the \(\Ff \)-operads.