The Stacks project

Theorem 7.50.2. Let $\mathcal{C}$ be a category. Let $J$, $J'$ be topologies on $\mathcal{C}$. The following are equivalent

  1. $J = J'$,

  2. sheaves for the topology $J$ are the same as sheaves for the topology $J'$.

Proof. It is a tautology that if $J = J'$ then the notions of sheaves are the same. Conversely, Lemma 7.50.1 characterizes covering sieves in terms of the sheafification functor. But the sheafification functor $\textit{PSh}(\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}, J)$ is the left adjoint of the inclusion functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}, J) \to \textit{PSh}(\mathcal{C})$. Hence if the subcategories $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}, J)$ and $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}, J')$ are the same, then the sheafification functors are the same and hence the collections of covering sieves are the same. $\square$

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