Trivial ∞-operads

Construction 14.1.18 (Trivial operads). Let \(C\) be an \(\infty \)-category. Applying Construction 14.1.16 to \(C\catop \) gives a cartesian fibration \(q\colon \Fin (C\catop ) \to \Fin \); passing to opposite \(\infty \)-categories, we obtain a functor \(q\catop \colon \Fin (C\catop )\catop \to \Fin \catop \). We define \[ \Triv _{C}^{\otimes } := \Fin (C\catop )\catop , \qquad \qquad p_{\Triv _C}\colon \Fin (C\catop )\catop \xrightarrow {q\catop } \Fin \catop \hookrightarrow \Span (\Fin ), \] and refer to \(\Triv _C\) as the trivial \(\infty \)-operad generated by \(C\). Unwinding Construction 14.1.16, the objects of \(\Triv _C^{\otimes }\) are finite unordered tuples \(\{x_i\}_{i \in I}\) of objects of \(C\), while a morphism \(\{x_i\}_{i \in I} \to \{y_j\}_{j \in J}\) lying over the backwards span \(I \xleftarrow {g} J \xrightarrow {=} J\) is a collection of morphisms \(x_{g(j)} \to y_j\) in \(C\). The opposite in the definition ensures that the underlying \(\infty \)-category is \(C\) itself, and not \(C\catop \).