Lemma 7.33.1. Let $\mathcal{C}$ be a site. Let $p = u : \mathcal{C} \to \textit{Sets}$ be a functor. If the category of neighbourhoods of $p$ is cofiltered, then the stalk functor (7.32.1.1) is left exact.
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.
Proof. 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 the opposite of the category of neighbourhoods is filtered by assumption. $\square$
Lemma 7.33.2. 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
$u(X)$ is a singleton set, and
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 the category of neighbourhoods of $p$ is cofiltered and consequently the stalk functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \to \mathcal{F}_ p$ commutes with finite limits.
Proof. Please note the analogy with Lemma 7.5.2. 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. The “consequently” clause is Lemma 7.33.1. $\square$
Proposition 7.33.3. 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
$u$ commutes with finite limits, and
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 7.33.2. $\square$
Remark 7.33.4. 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
$u(X) = \{ *\} $ a singleton, and
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.
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 7.33.5. 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 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.
\[ u : X_{Zar} \longrightarrow \textit{Sets}, \quad U \mapsto \left\{ \begin{matrix} \emptyset
& \text{if}
& x \not\in U
\\ \{ *\}
& \text{if}
& x \in U
\end{matrix} \right. \]
Example 7.33.6. 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 7.33.3. 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 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 Note that for any irreducible closed $Z \subset X$ this functor satisfies assumptions (1), (2) of Proposition 7.33.3 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 7.33.5.
\[ u(U) = \emptyset \Rightarrow \forall i, \ u(U_ i) = \emptyset \quad \text{and}\quad u(U) \not= \emptyset \Rightarrow \exists i, \ u(U_ i) \not= \emptyset . \]
\[ u : X_{Zar} \longrightarrow \textit{Sets}, \quad U \mapsto \left\{ \begin{matrix} \emptyset
& \text{if}
& Z \cap U = \emptyset
\\ \{ *\}
& \text{if}
& Z \cap U \not= \emptyset
\end{matrix} \right. \]
Example 7.33.7. 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 is the forgetful functor. The pushforward $p_*$ is the functor 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.
\[ p^{-1} : \mathop{\mathit{Sh}}\nolimits (\mathcal{T}_ G) = G\textit{-Sets} \longrightarrow \textit{Sets} \]
\[ \textit{Sets} \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{T}_ G) = G\textit{-Sets} \]
Example 7.33.8. 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 $u : U \mapsto \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U_0, U)$ 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.
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)