The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

7.34 Points and morphisms of topoi

In this section we make a few remarks about points and morphisms of topoi.

slogan

Lemma 7.34.1. Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites given by a continuous functor $u : \mathcal{C} \to \mathcal{D}$. Let $p$ be a point of $\mathcal{D}$ given by the functor $v : \mathcal{D} \to \textit{Sets}$, see Definition 7.32.2. Then the functor $v \circ u : \mathcal{C} \to \textit{Sets}$ defines a point $q$ of $\mathcal{C}$ and moreover there is a canonical identification

\[ (f^{-1}\mathcal{F})_ p = \mathcal{F}_ q \]

for any sheaf $\mathcal{F}$ on $\mathcal{C}$.

First proof Lemma 7.34.1. Note that since $u$ is continuous and since $v$ defines a point, it is immediate that $v \circ u$ satisfies conditions (1) and (2) of Definition 7.32.2. Let us prove the displayed equality. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. Then

\[ \mathcal{F}_ q = \mathop{\mathrm{colim}}\nolimits _{(U, x)} \mathcal{F}(U) \]

where the colimit is over objects $U$ in $\mathcal{C}$ and elements $x \in v(u(U))$. Similarly, we have

\begin{align*} (f^{-1}\mathcal{F})_ p & = (u_ p\mathcal{F})_ p \\ & = \mathop{\mathrm{colim}}\nolimits _{(V, x)} \mathop{\mathrm{colim}}\nolimits _{U, \phi : V \to u(U)} \mathcal{F}(U) \\ & = \mathop{\mathrm{colim}}\nolimits _{(V, x, U, \phi : V \to u(U))} \mathcal{F}(U) \\ & = \mathop{\mathrm{colim}}\nolimits _{(U, x)} \mathcal{F}(U) \\ & = \mathcal{F}_ q \end{align*}

Explanation: The first equality holds because $f^{-1}\mathcal{F} = (u_ p\mathcal{F})^\# $ and because $\mathcal{G}_ p = \mathcal{G}^\# _ p$ for any presheaf $\mathcal{G}$, see Lemma 7.32.5. The second equality holds by the definition of $u_ p$. In the third equality we simply combine colimits. To see the fourth equality we apply Categories, Lemma 4.17.5 to the functor $F$ of diagram categories defined by the rule $F((V, x, U, \phi : V \to u(U))) = (U, v(\phi )(x))$. The lemma applies, because $F$ has a right inverse, namely $(U, x) \mapsto (u(U), x, U, \text{id} : u(U) \to u(U))$ and because there is always a morphism

\[ (V, x, U, \phi : V \to u(U)) \longrightarrow (u(U), v(\phi )(x), U, \text{id} : u(U) \to u(U)) \]

in the fibre category over $(U, x)$ which shows the fibre categories are connected. The fifth equality is clear. Hence now we see that $q$ also satisfies condition (3) of Definition 7.32.2 because it is a composition of exact functors. This finishes the proof. $\square$

Second proof Lemma 7.34.1. By Lemma 7.32.8 we may factor $(p_*, p^{-1})$ as

\[ \mathop{\mathit{Sh}}\nolimits (pt) \xrightarrow {i} \mathop{\mathit{Sh}}\nolimits (\mathcal{S}) \xrightarrow {h} \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \]

where the second morphism of topoi comes from a morphism of sites $h : \mathcal{S} \to \mathcal{D}$ induced by the functor $v : \mathcal{D} \to \mathcal{S}$ (which makes sense as $\mathcal{S} \subset \textit{Sets}$ is a full subcategory containing every object in the image of $v$). By Lemma 7.14.4 the composition $v \circ u : \mathcal{C} \to \mathcal{S}$ defines a morphism of sites $g : \mathcal{S} \to \mathcal{C}$. In particular, the functor $v \circ u : \mathcal{C} \to \mathcal{S}$ is continuous which by the definition of the coverings in $\mathcal{S}$, see Remark 7.15.3, means that $v \circ u$ satisfies conditions (1) and (2) of Definition 7.32.2. On the other hand, we see that

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

by the construction of $i$ in Remark 7.15.3. Note that this is the same as the formula for which is equal to $(v \circ u)^ pE$, see Equation (7.32.3.1). By Lemma 7.32.5 the functor $g_*i_* = (v \circ u)^ p = (v \circ u)^ s$ is right adjoint to the stalk functor $\mathcal{F} \mapsto \mathcal{F}_ q$. Hence we see that the stalk functor $q^{-1}$ is canonically isomorphic to $i^{-1} \circ g^{-1}$. Hence it is exact and we conclude that $q$ is a point. Finally, as we have $g = f \circ h$ by construction we see that $q^{-1} = i^{-1} \circ h^{-1} \circ f^{-1} = p^{-1} \circ f^{-1}$, i.e., we have the displayed formula of the lemma. $\square$

Lemma 7.34.2. Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a morphism of topoi. Let $p : \mathop{\mathit{Sh}}\nolimits (pt) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be a point. Then $q = f \circ p$ is a point of the topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ and we have a canonical identification

\[ (f^{-1}\mathcal{F})_ p = \mathcal{F}_ q \]

for any sheaf $\mathcal{F}$ on $\mathcal{C}$.

Proof. This is immediate from the definitions and the fact that we can compose morphisms of topoi. $\square$


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