Čech nerve

Definition 5.1.9. Let \(C\) be an \(\infty \)-category with pullbacks. We define the Čech nerve functor \(\check {C}\colon \Ar (C) \to \Fun (\simp \catop ,C)\) as the composite \[ \check {C}_{\bullet }\colon \Ar (C) \simeq \Fun ([1]\catop ,C) \xrightarrow {j_*} \Fun (\simp _+\catop ,C) \xrightarrow {i^*} \Fun (\simp \catop ,C), \] where \(j_*\) is the right Kan extension functor along the inclusion \(j\catop \colon [1]\catop \hookrightarrow \simp _+\catop \), and where \(i^*\) is restriction along the inclusion \(i\catop \). Given a morphism \(f\colon X \to Y\) in \(C\), we refer to the simplicial object \(\check {C}_{\bullet }(f)\) as the Čech nerve of \(f\).