Effective epimorphism if and only if local sections

Suppose first that \(f\) admits local sections. If \(Y=y_{\tau}(Y')\), choose a covering family \(\{U_i\to Y'\}_{i\in I}\) over which \(f\) admits sections. The induced map is an effective epimorphism. After base change along this map, \(f\) is a coproduct of morphisms admitting sections and hence is an effective epimorphism by Lemma 2.38. Since effective epimorphisms form a local class, \(f\) is an effective epimorphism. For arbitrary \(Y\), the same argument applies after choosing an effective epimorphism from a coproduct of sheafified representables to \(Y\).Conversely, assume that \(f\colon X \to Y\) is an effective epimorphism. Since effective epimorphisms are stable under base change, it suffices to treat the case \(Y=y_{\tau}(Y')\). Let \(i\colon\Shv_{\tau}(C)\hookrightarrow\PSh(C)\) denote the inclusion and form the pullback along the unit \(y(Y')\to i(y_{\tau}(Y'))\). Consider the epi–mono factorization in \(\PSh(C)\): Applying sheafification \(L_{\tau}\colon \PSh(C)\to \Shv_{\tau}(C)\) gives a factorization where we have used left exactness of \(L_{\tau}\) to identify \(L_{\tau}(\widetilde X)\) with \(X\). The composite is \(f\), so right cancellation for effective epimorphisms shows that the map \(L_{\tau}(U_f)\to y_{\tau}(Y')\) is an effective epimorphism. It is also a monomorphism, since \(L_{\tau}\) is left exact, and hence it is an isomorphism.Therefore the monomorphism \(U_f\hookrightarrow y(Y')\) is inverted by \(L_{\tau}\). By Theorem 6.2, the class of morphisms inverted by \(L_{\tau}\) is the congruence \(\tau^c\). Thus \(U_f \hookrightarrow y(Y')\) lies in \(\tau^c \cap \Mono = \tau\) by Proposition 6.9, i.e. it is a covering sieve on \(Y'\). By construction, a morphism \(U\to Y'\) belongs to \(U_f\) precisely when the corresponding map \(y_{\tau}(U)\to y_{\tau}(Y')\) admits a lift to \(X\). Taking all morphisms in \(U_f\) as a covering family therefore shows that \(f\) admits local sections.