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*}

Remark 7.15.3. There are many sites that give rise to the topos $\mathop{\mathit{Sh}}\nolimits (pt)$. A useful example is the following. Suppose that $S$ is a set (of sets) which contains at least one nonempty element. Let $\mathcal{S}$ be the category whose objects are elements of $S$ and whose morphisms are arbitrary set maps. Assume that $\mathcal{S}$ has fibre products. For example this will be the case if $S = \mathcal{P}(\text{infinite set})$ is the power set of any infinite set (exercise in set theory). Make $\mathcal{S}$ into a site by declaring surjective families of maps to be coverings (and choose a suitable sufficiently large set of covering families as in Sets, Section 3.11). We claim that $\mathop{\mathit{Sh}}\nolimits (\mathcal{S})$ is equivalent to the category of sets.

We first prove this in case $S$ contains $e \in S$ which is a singleton. In this case, there is an equivalence of topoi $i : \mathop{\mathit{Sh}}\nolimits (pt) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{S})$ given by the functors

7.15.3.1
\begin{equation} \label{sites-equation-sheaves-pt-sets} i^{-1}\mathcal{F} = \mathcal{F}(e), \quad i_*E = (U \mapsto \mathop{Mor}\nolimits _{\textit{Sets}}(U, E)) \end{equation}

Namely, suppose that $\mathcal{F}$ is a sheaf on $\mathcal{S}$. For any $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}) = S$ we can find a covering $\{ \varphi _ u : e \to U\} _{u \in U}$, where $\varphi _ u$ maps the unique element of $e$ to $u \in U$. The sheaf condition implies in this case that $\mathcal{F}(U) = \prod _{u \in U} \mathcal{F}(e)$. In other words $\mathcal{F}(U) = \mathop{Mor}\nolimits _{\textit{Sets}}(U, \mathcal{F}(e))$. Moreover, this rule is compatible with restriction mappings. Hence the functor

\[ i_* : \textit{Sets} = \mathop{\mathit{Sh}}\nolimits (pt) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{S}), \quad E \longmapsto (U \mapsto \mathop{Mor}\nolimits _{\textit{Sets}}(U, E)) \]

is an equivalence of categories, and its inverse is the functor $i^{-1}$ given above.

If $\mathcal{S}$ does not contain a singleton, then the functor $i_*$ as defined above still makes sense. To show that it is still an equivalence in this case, choose any nonempty $\tilde e \in S$ and a map $\varphi : \tilde e \to \tilde e$ whose image is a singleton. For any sheaf $\mathcal{F}$ set

\[ \mathcal{F}(e) := \mathop{\mathrm{Im}}( \mathcal{F}(\varphi ) : \mathcal{F}(\tilde e) \longrightarrow \mathcal{F}(\tilde e) ) \]

and show that this is a quasi-inverse to $i_*$. Details omitted.


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 00XD. Beware of the difference between the letter 'O' and the digit '0'.