Definition 6.14. (Čech descent)
Let \(C\) be a category and let \(\Uu = \{U_i \to X\}_{i\in I}\) be a collection of morphisms. We will denote by \(\check{C}_{\bullet}(\Uu)\) the Čech nerve of the morphism
\[\bigsqcup_{i \in I} y(U_i) \to y(X)\]
in \(\PSh(C)\). A presheaf \(\Ff \in \PSh(C)\) is said to satisfy Čech descent with respect to \(\Uu\) if the map
\[\Ff(X) = \Hom_{\PSh(C)}(y(X),\Ff) \to \lim_{[n] \in \simp} \Hom_{\PSh(C)}(\check{C}_n(\Uu),\Ff)\]
is an equivalence. More concretely, if the relevant iterated fiber products of the \(U_i\) over \(X\) exist in \(C\), then \(\Ff\) satisfies Čech descent with respect to \(\Uu\) if the diagram
is a limit diagram.