Notation 6.13.
Let \(\Uu = \{U_i \to X\}_{i\in I}\) be a collection of morphisms in a category \(C\). The morphism \(\bigsqcup_{i \in I} y(U_i) \to y(X)\) in \(\PSh(C)\) factors as an effective epimorphism followed by a monomorphism:
\[\bigsqcup_{i \in I} y(U_i) \twoheadrightarrow U \hookrightarrow y(X).\]
We refer to the sieve \(U \hookrightarrow y(X)\) as the sieve generated by \(\Uu\).
Since limits and colimits in \(\PSh(C)\) are computed pointwise, we see that this epi-mono factorization is also computed pointwise: for every \(Z \in C\), the subanima
\[U(Z) \subseteq y(X)(Z) = \Hom_C(Z,X)\]
consists of those morphisms \(Z \to X\) in \(C\) that factor through \(U_i \to X\) for some \(i \in I\).
If \(C\) carries a Grothendieck topology \(\tau\), we say that \(\Uu\) is a covering family if the sieve it generates is a covering sieve.