The Stacks project

Lemma 7.35.2. Let $\mathcal{C}$, $p$, $u$, $U$ be as in Lemma 7.35.1. The construction of Lemma 7.35.1 gives a one to one correspondence between points $q$ of $\mathcal{C}/U$ lying over $p$ and elements $x$ of $u(U)$.

Proof. Let $q$ be a point of $\mathcal{C}/U$ given by the functor $v : \mathcal{C}/U \to \textit{Sets}$ such that $j_ U \circ q = p$ as morphisms of topoi. Recall that $u(V) = p^{-1}(h_ V^\# )$ for any object $V$ of $\mathcal{C}$, see Lemma 7.32.7. Similarly $v(V/U) = q^{-1}(h_{V/U}^\# )$ for any object $V/U$ of $\mathcal{C}/U$. Consider the following two diagrams

\[ \vcenter { \xymatrix{ \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}/U}(W/U, V/U) \ar[r] \ar[d] & \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, V) \ar[d] \\ \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}/U}(W/U, U/U) \ar[r] & \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, U) } } \quad \vcenter { \xymatrix{ h_{V/U}^\# \ar[r] \ar[d] & j_ U^{-1}(h_ V^\# ) \ar[d] \\ h_{U/U}^\# \ar[r] & j_ U^{-1}(h_ U^\# ) } } \]

The right hand diagram is the sheafification of the diagram of presheaves on $\mathcal{C}/U$ which maps $W/U$ to the left hand diagram of sets. (There is a small technical point to make here, namely, that we have $(j_ U^{-1}h_ V)^\# = j_ U^{-1}(h_ V^\# )$ and similarly for $h_ U$, see Lemma 7.20.4.) Note that the left hand diagram of sets is cartesian. Since sheafification is exact (Lemma 7.10.14) we conclude that the right hand diagram is cartesian.

Apply the exact functor $q^{-1}$ to the right hand diagram to get a cartesian diagram

\[ \xymatrix{ v(V/U) \ar[r] \ar[d] & u(V) \ar[d] \\ v(U/U) \ar[r] & u(U) } \]

of sets. Here we have used that $q^{-1} \circ j^{-1} = p^{-1}$. Since $U/U$ is a final object of $\mathcal{C}/U$ we see that $v(U/U)$ is a singleton. Hence the image of $v(U/U)$ in $u(U)$ is an element $x$, and the top horizontal map gives a bijection $v(V/U) \to \{ y \in u(V) \mid y \mapsto x\text{ in }u(U)\} $ as desired. $\square$

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