Grothendieck topologies on category and presheaf topos

If \(\tau\) is a Grothendieck topology on \(\PSh(C)\), we declare a sieve \(U \hookrightarrow y(X)\) to be covering if it is a covering monomorphism in \(\PSh(C)\), i.e. if it belongs to \(\tau\). Conditions (1) and (2) follow from (a) and base-change stability. For (3), let \(i\colon U \hookrightarrow y(X)\) be a covering sieve and let \(j\colon V \hookrightarrow y(X)\) be a sieve such that for every map \(f\colon y(Y) \to U\) from a representable the projection \(y(Y) \times_{y(X)} V \hookrightarrow y(Y)\) is a covering monomorphism. Since \(\tau\) is local, it follows by descent that \(U \times_{y(X)} V \to U\) is a covering monomorphism. Its composite with \(U \hookrightarrow y(X)\) is covering by (c), and hence \(V \hookrightarrow y(X)\) is covering by (d).Conversely, given a Grothendieck topology on \(C\), we define a monomorphism \(U \hookrightarrow X\) in \(\PSh(C)\) to be covering if its pullback along every map \(y(Y) \to X\) from a representable is a covering sieve. We verify properties (a)–(d).For (a), all isomorphisms are covering. Moreover, the saturation \(\tau^s\) is generated by the set of covering sieves on representables. Indeed, every monomorphism \(U \hookrightarrow X\) is the colimit in \(\Ar(\PSh(C))\) of its pullbacks along maps \(y(Y) \to X\), by the density of the Yoneda embedding. Hence every covering monomorphism belongs to the saturation of the covering sieves.For (b), stability under base change follows directly from the definition, and closure under coproducts follows because every map from a representable into a coproduct factors through one of its summands. For descent, consider a pullback square as in Definition 2.46 in which the bottom map is an effective epimorphism and the upper vertical map is covering. After pulling back along a map from a representable, the bottom effective epimorphism admits a section: this follows by evaluating it at the object representing its target. The corresponding pullback of the lower vertical map is therefore a base change of the upper vertical map and hence is covering.For (c), let \(V \hookrightarrow U\) and \(U \hookrightarrow X\) be covering monomorphisms. To show that \(V \hookrightarrow X\) is covering, we may pull back along a map from a representable and thus assume that \(X = y(X')\). Then \(U \hookrightarrow y(X')\) is a covering sieve. By axiom (3), it suffices to show that for every \(f\colon Y \to X'\) in \(U\), the pullback sieve \(f^*V \hookrightarrow y(Y)\) is covering. This map is a base change of \(V \hookrightarrow U\), so it is covering by assumption.For (d), consider monomorphisms \(V \xhookrightarrow{f} U \xhookrightarrow{g} X\) such that \(gf \in \tau\). We may again assume that \(X = y(X')\) is representable. Then \(V \hookrightarrow y(X')\) is a covering sieve. By axiom (3), it suffices to show that for every \(h\colon Y \to X'\) in \(V\), the pullback \(h^*U \hookrightarrow y(Y)\) is covering. Since \(h\) factors through \(V \hookrightarrow U\), this pullback is the identity of \(y(Y)\) and hence is covering.It is immediate from the construction that these two assignments are inverse to each other, finishing the proof.