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.


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 ( 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. $\square$

Comments (0)

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05V0. Beware of the difference between the letter 'O' and the digit '0'.