Proposition 6.15. ({cf. [Lurie 2009, Lemma 6.2.3.18]})
Let \(\Uu = \{U_i \to X\}_{i\in I}\) be a collection of morphisms in a category \(C\) and let \(U \hookrightarrow y(X)\) be the sieve generated by \(\Uu\). Then a presheaf \(\Ff\) on \(C\) satisfies Čech descent with respect to \(\Uu\) if and only if it is local with respect to \(U \hookrightarrow y(X)\).
Proof
By definition, \(U \hookrightarrow y(X)\) is obtained from the epi-mono factorization \(\bigsqcup_{i \in I} y(U_i) \twoheadrightarrow U \hookrightarrow y(X)\). It follows that \(U\) is equivalent to the colimit of the Čech nerve \(\check{C}(\Uu)\) of \(\Uu\), and we obtain for every \(\Ff \in \PSh(C)\) an equivalence
\[\Hom_{\PSh(C)}(U,\Ff) \simeq \Hom_{\PSh(C)}(\colim_{[n] \in \simp\catop} \check{C}_n(\Uu),\Ff) \simeq \lim_{[n] \in \simp}\Hom_{\PSh(C)}(\check{C}_n(\Uu),\Ff).\]
The claim now follows immediately.References
- Jacob Lurie. Higher topos theory. Ann. Math. Stud. 170, Princeton, NJ: Princeton University Press. 2009.