Trivial operad special cases

Example 14.1.22. Taking \(C = *\) recovers the trivial \(\infty \)-operad \[ \Triv := \Triv _*, \qquad \Triv ^{\otimes } = \Fin \catop , \qquad p_{\Triv } = \incl \colon \Fin \catop \hookrightarrow \Span (\Fin ). \] Taking \(C = \emptyset \) gives the empty \(\infty \)-operad \(\Triv _{\emptyset }\), with \(\Triv _{\emptyset }^{\otimes } = *\) and \(p_{\Triv _{\emptyset }}\colon * \xhookrightarrow {\lra {0}} \Span (\Fin )\) the inclusion of the empty set \(\lra {0}\). Since \(\Triv _{(-)}\) is a left adjoint by Corollary 14.1.21, it preserves initial objects; as \(\emptyset \) is initial in \(\Cat _{\infty }\), we conclude that \(\Triv _{\emptyset }\) is the initial \(\infty \)-operad.