The Stacks project

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
\begin{equation} \label{sites-equation-sheaves-pt-sets} i^{-1}\mathcal{F} = \mathcal{F}(e), \quad i_*E = (U \mapsto \mathop{\mathrm{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 $\mathcal{F}(U) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Sets}}(U, \mathcal{F}(e))$. Moreover, this rule is compatible with restriction mappings. Hence the functor

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

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.

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