Definition 7.32.1. Let $\mathcal{C}$ be a site. A *point of the topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$* is a morphism of topoi $p$ from $\mathop{\mathit{Sh}}\nolimits (pt)$ to $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

## 7.32 Points

We will define a point of a site in terms of a functor $u : \mathcal{C} \to \textit{Sets}$. It will turn out later that $u$ will define a morphism of sites which gives rise to a point of the topos associated to $\mathcal{C}$, see Lemma 7.32.8.

Let $\mathcal{C}$ be a site. Let $p = u$ be a functor $u : \mathcal{C} \to \textit{Sets}$. This curious language is introduced because it seems funny to talk about neighbourhoods of functors; so we think of a “point” $p$ as a geometric thing which is given by a categorical datum, namely the functor $u$. The fact that $p$ is actually equal to $u$ does not matter. A *neighbourhood* of $p$ is a pair $(U, x)$ with $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and $x \in u(U)$. A *morphism of neighbourhoods* $(V, y) \to (U, x)$ is given by a morphism $\alpha :V \to U$ of $\mathcal{C}$ such that $u(\alpha )(y) = x$. Note that the category of neighbourhoods isn't a “big” category.

We define the *stalk* of a presheaf $\mathcal{F}$ at $p$ as

The colimit is over the opposite of the category of neighbourhoods of $p$. In other words, an element of $\mathcal{F}_ p$ is given by a triple $(U, x, s)$, where $(U, x)$ is a neighbourhood of $p$ and $s \in \mathcal{F}(U)$. Equality of triples is the equivalence relation generated by $(U, x, s) \sim (V, y, \alpha ^*s)$ when $\alpha $ is as above.

Note that if $\varphi : \mathcal{F} \to \mathcal{G}$ is a morphism of presheaves of sets, then we get a canonical map of stalks $\varphi _ p : \mathcal{F}_ p \to \mathcal{G}_ p$. Thus we obtain a *stalk functor*

We have defined the stalk functor using any functor $p = u : \mathcal{C} \to \textit{Sets}$. No conditions are necessary for the definition to work^{1}. On the other hand, it is probably better not to use this notion unless $p$ actually is a point (see definition below), since in general the stalk functor does not have good properties.

Definition 7.32.2. Let $\mathcal{C}$ be a site. A *point $p$ of the site $\mathcal{C}$* is given by a functor $u : \mathcal{C} \to \textit{Sets}$ such that

For every covering $\{ U_ i \to U\} $ of $\mathcal{C}$ the map $\coprod u(U_ i) \to u(U)$ is surjective.

For every covering $\{ U_ i \to U\} $ of $\mathcal{C}$ and every morphism $V \to U$ the maps $u(U_ i \times _ U V) \to u(U_ i) \times _{u(U)} u(V)$ are bijective.

The stalk functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \mapsto \mathcal{F}_ p$ is left exact.

The conditions should be familiar since they are modeled after those of Definitions 7.13.1 and 7.14.1. Note that (3) implies that $*_ p = \{ *\} $, see Example 7.10.2. Hence $u(U) \not= \emptyset $ for at least some $U$ (because the empty colimit produces the empty set). We will show below (Lemma 7.32.7) that this does give rise to a point of the topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$. Before we do so, we prove some lemmas for general functors $u$.

Lemma 7.32.3. Let $\mathcal{C}$ be a site. Let $p = u : \mathcal{C} \to \textit{Sets}$ be a functor. There are functorial isomorphisms $(h_ U)_ p = u(U)$ for $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

**Proof.**
An element of $(h_ U)_ p$ is given by a triple $(V, y, f)$, where $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, $y\in u(V)$ and $f \in h_ U(V) = \mathop{Mor}\nolimits _\mathcal {C}(V, U)$. Two such $(V, y, f)$, $(V', y', f')$ determine the same object if there exists a morphism $\phi : V \to V'$ such that $u(\phi )(y) = y'$ and $f' \circ \phi = f$, and in general you have to take chains of identities like this to get the correct equivalence relation. In any case, every $(V, y, f)$ is equivalent to the element $(U, u(f)(y), \text{id}_ U)$. If $\phi $ exists as above, then the triples $(V, y, f)$, $(V', y', f')$ determine the same triple $(U, u(f)(y), \text{id}_ U) = (U, u(f')(y'), \text{id}_ U)$. This proves that the map $u(U) \to (h_ U)_ p$, $x \mapsto \text{class of }(U, x, \text{id}_ U)$ is bijective.
$\square$

Let $\mathcal{C}$ be a site. Let $p = u : \mathcal{C} \to \textit{Sets}$ be a functor. In analogy with the constructions in Section 7.5 given a set $E$ we define a presheaf $u^ pE$ by the rule

This defines a functor $u^ p : \textit{Sets} \to \textit{PSh}(\mathcal{C})$, $E \mapsto u^ pE$.

Lemma 7.32.4. For any functor $u : \mathcal{C} \to \textit{Sets}$. The functor $u^ p$ is a right adjoint to the stalk functor on presheaves.

**Proof.**
Let $\mathcal{F}$ be a presheaf on $\mathcal{C}$. Let $E$ be a set. A morphism $\mathcal{F} \to u^ pE$ is given by a compatible system of maps $\mathcal{F}(U) \to \text{Map}(u(U), E)$, i.e., a compatible system of maps $\mathcal{F}(U) \times u(U) \to E$. And by definition of $\mathcal{F}_ p$ a map $\mathcal{F}_ p \to E$ is given by a rule associating with each triple $(U, x, \sigma )$ an element in $E$ such that equivalent triples map to the same element, see discussion surrounding Equation (7.32.1.1). This also means a compatible system of maps $\mathcal{F}(U) \times u(U) \to E$.
$\square$

In analogy with Section 7.13 we have the following lemma.

Lemma 7.32.5. Let $\mathcal{C}$ be a site. Let $p = u : \mathcal{C} \to \textit{Sets}$ be a functor. Suppose that for every covering $\{ U_ i \to U\} $ of $\mathcal{C}$

the map $\coprod u(U_ i) \to u(U)$ is surjective, and

the maps $u(U_ i \times _ U U_ j) \to u(U_ i) \times _{u(U)} u(U_ j)$ are surjective.

Then we have

the presheaf $u^ pE$ is a sheaf for all sets $E$, denote it $u^ sE$,

the stalk functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \textit{Sets}$ and the functor $u^ s: \textit{Sets} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ are adjoint, and

we have $\mathcal{F}_ p = \mathcal{F}^\# _ p$ for every presheaf of sets $\mathcal{F}$.

**Proof.**
The first assertion is immediate from the definition of a sheaf, assumptions (1) and (2), and the definition of $u^ pE$. The second is a restatement of the adjointness of $u^ p$ and the stalk functor (Lemma 7.32.4) restricted to sheaves. The third assertion follows as, for any set $E$, we have

by the adjointness property of sheafification. $\square$

In particular Lemma 7.32.5 holds when $p = u$ is a point. In this case we think of the sheaf $u^ sE$ as the “skyscraper” sheaf with value $E$ at $p$.

Definition 7.32.6. Let $p$ be a point of the site $\mathcal{C}$ given by the functor $u$. For a set $E$ we define $p_*E = u^ sE$ the sheaf described in Lemma 7.32.5 above. We sometimes call this a *skyscraper sheaf*.

In particular we have the following adjointness property of skyscraper sheaves and stalks:

This motivates the notation $p^{-1}\mathcal{F} = \mathcal{F}_ p$ which we will sometimes use.

Lemma 7.32.7. Let $\mathcal{C}$ be a site.

Let $p$ be a point of the site $\mathcal{C}$. Then the pair of functors $(p_*, p^{-1})$ introduced above define a morphism of topoi $\mathop{\mathit{Sh}}\nolimits (pt) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

Let $p = (p_*, p^{-1})$ be a point of the topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$. Then the functor $u : U \mapsto p^{-1}(h_ U^\# )$ gives rise to a point $p'$ of the site $\mathcal{C}$ whose associated morphism of topoi $(p'_*, (p')^{-1})$ is equal to $p$.

**Proof.**
Proof of (1). By the above the functors $p_*$ and $p^{-1}$ are adjoint. The functor $p^{-1}$ is required to be exact by Definition 7.32.2. Hence the conditions imposed in Definition 7.15.1 are all satisfied and we see that (1) holds.

Proof of (2). Let $\{ U_ i \to U\} $ be a covering of $\mathcal{C}$. Then $\coprod (h_{U_ i})^\# \to h_ U^\# $ is surjective, see Lemma 7.12.4. Since $p^{-1}$ is exact (by definition of a morphism of topoi) we conclude that $\coprod u(U_ i) \to u(U)$ is surjective. This proves part (1) of Definition 7.32.2. Sheafification is exact, see Lemma 7.10.14. Hence if $U \times _ V W$ exists in $\mathcal{C}$, then

and we see that $u(U \times _ V W) = u(U) \times _{u(V)} u(W)$ since $p^{-1}$ is exact. This proves part (2) of Definition 7.32.2. Let $p' = u$, and let $\mathcal{F}_{p'}$ be the stalk functor defined by Equation (7.32.1.1) using $u$. There is a canonical comparison map $c : \mathcal{F}_{p'} \to \mathcal{F}_ p = p^{-1}\mathcal{F}$. Namely, given a triple $(U, x, \sigma )$ representing an element $\xi $ of $\mathcal{F}_{p'}$ we think of $\sigma $ as a map $\sigma : h_ U^\# \to \mathcal{F}$ and we can set $c(\xi ) = p^{-1}(\sigma )(x)$ since $x \in u(U) = p^{-1}(h_ U^\# )$. By Lemma 7.32.3 we see that $(h_ U)_{p'} = u(U)$. Since conditions (1) and (2) of Definition 7.32.2 hold for $p'$ we also have $(h_ U^\# )_{p'} = (h_ U)_{p'}$ by Lemma 7.32.5. Hence we have

We claim this bijection equals the comparison map $c : (h_ U^\# )_{p'} \to p^{-1}(h_ U^\# )$ (verification omitted). Any sheaf on $\mathcal{C}$ is a coequalizer of maps of coproducts of sheaves of the form $h_ U^\# $, see Lemma 7.12.5. The stalk functor $\mathcal{F} \mapsto \mathcal{F}_{p'}$ and the functor $p^{-1}$ commute with arbitrary colimits (as they are both left adjoints). We conclude $c$ is an isomorphism for every sheaf $\mathcal{F}$. Thus the stalk functor $\mathcal{F} \mapsto \mathcal{F}_{p'}$ is isomorphic to $p^{-1}$ and we in particular see that it is exact. This proves condition (3) of Definition 7.32.2 holds and $p'$ is a point. The final assertion has already been shown above, since we saw that $p^{-1} = (p')^{-1}$. $\square$

Actually a point always corresponds to a morphism of sites as we show in the following lemma.

Lemma 7.32.8. Let $\mathcal{C}$ be a site. Let $p$ be a point of $\mathcal{C}$ given by $u : \mathcal{C} \to \textit{Sets}$. Let $S_0$ be an infinite set such that $u(U) \subset S_0$ for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $\mathcal{S}$ be the site constructed out of the powerset $S = \mathcal{P}(S_0)$ in Remark 7.15.3. Then

there is an equivalence $i : \mathop{\mathit{Sh}}\nolimits (pt) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{S})$,

the functor $u : \mathcal{C} \to \mathcal{S}$ induces a morphism of sites $f : \mathcal{S} \to \mathcal{C}$, and

the composition

\[ \mathop{\mathit{Sh}}\nolimits (pt) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{S}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \]is the morphism of topoi $(p_*, p^{-1})$ of Lemma 7.32.7.

**Proof.**
Part (1) we saw in Remark 7.15.3. Moreover, recall that the equivalence associates to the set $E$ the sheaf $i_*E$ on $\mathcal{S}$ defined by the rule $V \mapsto \mathop{Mor}\nolimits _{\textit{Sets}}(V, E)$. Part (2) is clear from the definition of a point of $\mathcal{C}$ (Definition 7.32.2) and the definition of a morphism of sites (Definition 7.14.1). Finally, consider $f_*i_*E$. By construction we have

which is equal to $p_*E(U)$, see Equation (7.32.3.1). This proves (3). $\square$

Contrary to what happens in the topological case it is not always true that the stalk of the skyscraper sheaf with value $E$ is $E$. Here is what is true in general.

Lemma 7.32.9. 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 $\mathcal{C}$. For any set $E$ there are canonical maps

whose composition is $\text{id}_ E$.

**Proof.**
There is always an adjunction map $(p_*E)_ p = p^{-1}p_*E \to E$. This map is an isomorphism when $E = \{ *\} $ because $p_*$ and $p^{-1}$ are both left exact, hence transform the final object into the final object. Hence given $e \in E$ we can consider the map $i_ e : \{ *\} \to E$ which gives

whence the map $E \to (p_*E)_ p = p^{-1}p_*E$ as desired. $\square$

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

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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