The Stacks project

Lemma 7.32.10. Let $\mathcal{C}$ be a site. Let $p : \mathop{\mathit{Sh}}\nolimits (pt) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a point of the topos associated to $\mathcal{C}$. The functor $p_* : \textit{Sets} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ has the following properties: It commutes with arbitrary limits, it is left exact, it is faithful, it transforms surjections into surjections, it commutes with coequalizers, it reflects injections, it reflects surjections, and it reflects isomorphisms.

Proof. Because $p_*$ is a right adjoint it commutes with arbitrary limits and it is left exact. The fact that $p^{-1}p_*E \to E$ is a canonically split surjection implies that $p_*$ is faithful, reflects injections, reflects surjections, and reflects isomorphisms. By Lemma 7.32.7 we may assume that $p$ comes from a point $u : \mathcal{C} \to \textit{Sets}$ of the underlying site $\mathcal{C}$. In this case the sheaf $p_*E$ is given by

\[ p_*E(U) = \mathop{Mor}\nolimits _{\textit{Sets}}(u(U), E) \]

see Equation (7.32.3.1) and Definition 7.32.6. It follows immediately from this formula that $p_*$ transforms surjections into surjections and coequalizers into coequalizers. $\square$


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