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
7.32.1.1
\begin{equation} \label{sites-equation-stalk} \mathcal{F}_ p = \mathop{\mathrm{colim}}\nolimits _{\{ (U, x)\} ^{opp}} \mathcal{F}(U). \end{equation}
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
\[ \textit{PSh}(\mathcal{C}) \longrightarrow \textit{Sets}, \quad \mathcal{F} \longmapsto \mathcal{F}_ p. \]
We have defined the stalk functor using any functor $p = u : \mathcal{C} \to \textit{Sets}$. No conditions are necessary for the definition to work1. 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 $f \in h_ U(V) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U)$. 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
7.32.3.1
\begin{equation} \label{sites-equation-skyscraper} U \longmapsto u^ pE(U) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Sets}}(u(U), E) = \text{Map}(u(U), E). \end{equation}
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
\[ \text{Map}(\mathcal{F}_ p, E) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(\mathcal{F}, u^ pE) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{F}^\# , u^ sE) = \text{Map}(\mathcal{F}^\# _ p, E) \]
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:
\[ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{F}, p_*E) = \text{Map}(\mathcal{F}_ p, E) \]
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
\[ h_{U \times _ V W}^\# = h_ U^\# \times _{h_ V^\# } h_ W^\# \]
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
\[ (h_ U^\# )_{p'} = (h_ U)_{p'} = u(U) = p^{-1}(h_ U^\# ) \]
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
is the morphism of topoi $(p_*, p^{-1})$ of Lemma 7.32.7.
\[ \mathop{\mathit{Sh}}\nolimits (pt) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{S}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \]
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 $V \mapsto \mathop{\mathrm{Mor}}\nolimits _{\textit{Sets}}(V, E)$. 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
\[ f_*i_*E(U) = i_*E(u(U)) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Sets}}(u(U), E) \]
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 $\text{id}_ E$.
\[ E \longrightarrow (p_*E)_ p \longrightarrow E \]
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
\[ \xymatrix{ p^{-1}p_*\{ *\} \ar[rr]_{p^{-1}p_*i_ e} \ar[d]_{\cong } & & p^{-1}p_*E \ar[d] \\ \{ *\} \ar[rr]^{i_ e} & & E } \]
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
\[ p_*E(U) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Sets}}(u(U), E) \]
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. $\square$
[1] One should try to avoid the case where $u(U) = \emptyset $ for all $U$.
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