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*}

7.33 Constructing points

In this section we give criteria for when a functor from a site to the category of sets defines a point of that site.

Lemma 7.33.1. Let $\mathcal{C}$ be a site. Assume that $\mathcal{C}$ has a final object $X$ and fibred products. Let $p = u : \mathcal{C} \to \textit{Sets}$ be a functor such that

  1. $u(X)$ is a singleton set, and

  2. for every pair of morphisms $U \to W$ and $V \to W$ with the same target the map $u(U \times _ W V) \to u(U) \times _{u(W)} u(V)$ is bijective.

Then the opposite of the category of neighbourhoods of $p$ is filtered. Moreover, the stalk functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \to \mathcal{F}_ p$ commutes with finite limits.

Proof. This is analogous to the proof of Lemma 7.5.2 above. The assumptions on $\mathcal{C}$ imply that $\mathcal{C}$ has finite limits. See Categories, Lemma 4.18.4. Assumption (1) implies that the category of neighbourhoods is nonempty. Suppose $(U, x)$ and $(V, y)$ are neighbourhoods. Then $u(U \times V) = u(U \times _ X V) = u(U) \times _{u(X)} u(V) = u(U) \times u(V)$ by (2). Hence there exists a neighbourhood $(U \times _ X V, z)$ mapping to both $(U, x)$ and $(V, y)$. Let $a, b : (V, y) \to (U, x)$ be two morphisms in the category of neighbourhoods. Let $W$ be the equalizer of $a, b : V \to U$. As in the proof of Categories, Lemma 4.18.4 we may write $W$ in terms of fibre products:

\[ W = (V \times _{a, U, b} V) \times _{(pr_1, pr_2), V \times V, \Delta } V \]

The bijectivity in (2) guarantees there exists an element $z \in u(W)$ which maps to $((y, y), y)$. Then $(W, z) \to (V, y)$ equalizes $a, b$ as desired.

Let $\mathcal{I} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$, $i \mapsto \mathcal{F}_ i$ be a finite diagram of sheaves. We have to show that the stalk of the limit of this system agrees with the limit of the stalks. Let $\mathcal{F}$ be the limit of the system as a presheaf. According to Lemma 7.10.1 this is a sheaf and it is the limit in the category of sheaves. Hence we have to show that $\mathcal{F}_ p = \mathop{\mathrm{lim}}\nolimits _\mathcal {I} \mathcal{F}_{i, p}$. Recall also that $\mathcal{F}$ has a simple description, see Section 7.4. Thus we have to show that

\[ \mathop{\mathrm{lim}}\nolimits _ i \mathop{\mathrm{colim}}\nolimits _{\{ (U, x)\} ^{opp}} \mathcal{F}_ i(U) = \mathop{\mathrm{colim}}\nolimits _{\{ (U, x)\} ^{opp}} \mathop{\mathrm{lim}}\nolimits _ i \mathcal{F}_ i(U). \]

This holds, by Categories, Lemma 4.19.2, because we just showed the opposite of the category of neighbourhoods is filtered. $\square$

Proposition 7.33.2. Let $\mathcal{C}$ be a site. Assume that finite limits exist in $\mathcal{C}$. (I.e., $\mathcal{C}$ has fibre products, and a final object.) A point $p$ of such a site $\mathcal{C}$ is given by a functor $u : \mathcal{C} \to \textit{Sets}$ such that

  1. $u$ commutes with finite limits, and

  2. if $\{ U_ i \to U\} $ is a covering, then $\coprod _ i u(U_ i) \to u(U)$ is surjective.

Proof. Suppose first that $p$ is a point (Definition 7.32.2) given by a functor $u$. Condition (2) is satisfied directly from the definition of a point. By Lemma 7.32.3 we have $(h_ U)_ p = u(U)$. By Lemma 7.32.5 we have $(h_ U^\# )_ p = (h_ U)_ p$. Thus we see that $u$ is equal to the composition of functors

\[ \mathcal{C} \xrightarrow {h} \textit{PSh}(\mathcal{C}) \xrightarrow {{}^\# } \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \xrightarrow {()_ p} \textit{Sets} \]

Each of these functors is left exact, and hence we see $u$ satisfies (1).

Conversely, suppose that $u$ satisfies (1) and (2). In this case we immediately see that $u$ satisfies the first two conditions of Definition 7.32.2. And its stalk functor is exact, because it is a left adjoint by Lemma 7.32.5 and it commutes with finite limits by Lemma 7.33.1. $\square$

Remark 7.33.3. In fact, let $\mathcal{C}$ be a site. Assume $\mathcal{C}$ has a final object $X$ and fibre products. Let $p = u: \mathcal{C} \to \textit{Sets}$ be a functor such that

  1. $u(X) = \{ *\} $ a singleton, and

  2. for every pair of morphisms $U \to W$ and $V \to W$ with the same target the map $u(U \times _ W V) \to u(U) \times _{u(W)} u(V)$ is surjective.

  3. for every covering $\{ U_ i \to U\} $ the map $\coprod u(U_ i) \to u(U)$ is surjective.

Then, in general, $p$ is not a point of $\mathcal{C}$. An example is the category $\mathcal{C}$ with two objects $\{ U, X\} $ and exactly one non-identity arrow, namely $U \to X$. We endow $\mathcal{C}$ with the trivial topology, i.e., the only coverings are $\{ U \to U\} $ and $\{ X \to X\} $. A sheaf $\mathcal{F}$ is the same thing as a presheaf and consists of a triple $(A, B, A \to B)$: namely $A = \mathcal{F}(X)$, $B = \mathcal{F}(U)$ and $A \to B$ is the restriction mapping corresponding to $U \to X$. Note that $U \times _ X U = U$ so fibre products exist. Consider the functor $u = p$ with $u(X) = \{ *\} $ and $u(U) = \{ *_1, *_2\} $. This satisfies (1), (2), and (3), but the corresponding stalk functor (7.32.1.1) is the functor

\[ (A, B, A \to B) \longmapsto B \amalg _ A B \]

which isn't exact. Namely, consider $(\emptyset , \{ 1\} , \emptyset \to \{ 1\} ) \to (\{ 1\} , \{ 1\} , \{ 1\} \to \{ 1\} )$ which is an injective map of sheaves, but is transformed into the noninjective map of sets

\[ \{ 1\} \amalg \{ 1\} \longrightarrow \{ 1\} \amalg _{\{ 1\} } \{ 1\} \]

by the stalk functor.

Example 7.33.4. Let $X$ be a topological space. Let $X_{Zar}$ be the site of Example 7.6.4. Let $x \in X$ be a point. Consider the functor

\[ u : X_{Zar} \longrightarrow \textit{Sets}, \quad U \mapsto \left\{ \begin{matrix} \emptyset & \text{if} & x \not\in U \\ \{ *\} & \text{if} & x \in U \end{matrix} \right. \]

This functor commutes with product and fibred products, and turns coverings into surjective families of maps. Hence we obtain a point $p$ of the site $X_{Zar}$. It is immediately verified that the stalk functor agrees with the stalk at $x$ defined in Sheaves, Section 6.11.

Example 7.33.5. Let $X$ be a topological space. What are the points of the topos $\mathop{\mathit{Sh}}\nolimits (X)$? To see this, let $X_{Zar}$ be the site of Example 7.6.4. By Lemma 7.32.7 a point of $\mathop{\mathit{Sh}}\nolimits (X)$ corresponds to a point of this site. Let $p$ be a point of the site $X_{Zar}$ given by the functor $u : X_{Zar} \to \textit{Sets}$. We are going to use the characterization of such a $u$ in Proposition 7.33.2. This implies immediately that $u(\emptyset ) = \emptyset $ and $u(U \cap V) = u(U) \times u(V)$. In particular we have $u(U) = u(U) \times u(U)$ via the diagonal map which implies that $u(U)$ is either a singleton or empty. Moreover, if $U = \bigcup U_ i$ is an open covering then

\[ u(U) = \emptyset \Rightarrow \forall i, \ u(U_ i) = \emptyset \quad \text{and}\quad u(U) \not= \emptyset \Rightarrow \exists i, \ u(U_ i) \not= \emptyset . \]

We conclude that there is a unique largest open $W \subset X$ with $u(W) = \emptyset $, namely the union of all the opens $U$ with $u(U) = \emptyset $. Let $Z = X \setminus W$. If $Z = Z_1 \cup Z_2$ with $Z_ i \subset Z$ closed, then $W = (X \setminus Z_1) \cap (X \setminus Z_2)$ so $\emptyset = u(W) = u(X \setminus Z_1) \times u(X \setminus Z_2)$ and we conclude that $u(X \setminus Z_1) = \emptyset $ or that $u(X \setminus Z_2) = \emptyset $. This means that $X \setminus Z_1 = W$ or that $X \setminus Z_2 = W$. In other words, $Z$ is irreducible. Now we see that $u$ is described by the rule

\[ u : X_{Zar} \longrightarrow \textit{Sets}, \quad U \mapsto \left\{ \begin{matrix} \emptyset & \text{if} & Z \cap U = \emptyset \\ \{ *\} & \text{if} & Z \cap U \not= \emptyset \end{matrix} \right. \]

Note that for any irreducible closed $Z \subset X$ this functor satisfies assumptions (1), (2) of Proposition 7.33.2 and hence defines a point. In other words we see that points of the site $X_{Zar}$ are in one-to-one correspondence with irreducible closed subsets of $X$. In particular, if $X$ is a sober topological space, then points of $X_{Zar}$ and points of $X$ are in one to one correspondence, see Example 7.33.4.

Example 7.33.6. Consider the site $\mathcal{T}_ G$ described in Example 7.6.5 and Section 7.9. The forgetful functor $u : \mathcal{T}_ G \to \textit{Sets}$ commutes with products and fibred products and turns coverings into surjective families. Hence it defines a point of $\mathcal{T}_ G$. We identify $\mathop{\mathit{Sh}}\nolimits (\mathcal{T}_ G)$ and $G\textit{-Sets}$. The stalk functor

\[ p^{-1} : \mathop{\mathit{Sh}}\nolimits (\mathcal{T}_ G) = G\textit{-Sets} \longrightarrow \textit{Sets} \]

is the forgetful functor. The pushforward $p_*$ is the functor

\[ \textit{Sets} \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{T}_ G) = G\textit{-Sets} \]

which maps a set $S$ to the $G$-set $\text{Map}(G, S)$ with action $g \cdot \psi = \psi \circ R_ g$ where $R_ g$ is right multiplication. In particular we have $p^{-1}p_*S = \text{Map}(G, S)$ as a set and the maps $S \to \text{Map}(G, S) \to S$ of Lemma 7.32.9 are the obvious ones.

Example 7.33.7. Let $\mathcal{C}$ be a category endowed with the chaotic topology (Example 7.6.6). For every object $U_0$ of $\mathcal{C}$ the functor $u : U \mapsto \mathop{Mor}\nolimits _\mathcal {C}(U_0, U)$ defines a point $p$ of $\mathcal{C}$. Namely, conditions (1) and (2) of Definition 7.32.2 are immediate as the only coverings are given by identity maps. Condition (2) holds because $\mathcal{F}_ p = \mathcal{F}(U_0)$ and since the topology is discrete taking sections over $U_0$ is an exact functor.


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