Does every $\kappa$-compact topos embedd relatively $\kappa$-tidily into a presheaf topos?

Question

Let $\kappa$ be a regular cardinal and say that a topos $\mathcal{E}$ is $\kappa$-compact if the global sections $\gamma_{\ast} : \mathcal{E} \to \mathsf{Set}$ preserves $\kappa$-filtered colimits.

My question is whether the following statement is true:

For every $\kappa$-compact topos $\mathcal{E}$, there is a finitely complete category $\mathcal{C}$ with an embedding $i : \mathcal{E} \hookrightarrow \mathsf{Psh}(\mathcal{C})$ for which the inclusion $i_{\ast}$ preserves $\kappa$-filtered colimits.

In other words, is there a presentation of any $\kappa$-compact $\mathcal{E}$ as sheaves on a finitely complete site $(\mathcal{C}, j)$ such that the $\kappa$-filtered colimit of sheaves is a sheaf?

If $\kappa = \omega$, the result is true for coherent $\mathcal{E}$ by taking a coherent site presentation of $\mathcal{E}$; since the covering families are finite, the sheaf condition becomes a finite limit and finite limits commute with filtered colimits.

Note that the global sections functor of $\mathsf{Psh}(\mathcal{C})$ will commute with all colimits, since it is given by evaluation at the terminal object of $\mathcal{C}$ and colimits in presheaves are computed component-wise. Therefore, if $(\mathcal{C}, j)$ is a site for which the $\kappa$-filtered colimit of sheaves is a sheaf, then $\mathsf{Sh} (\mathcal{C}, j)$ will be $\kappa$-compact. The question can therefore also be understood as asking whether this class of toposes is precisely the $\kappa$-compact toposes.

I'm interested in higher categorical answers as well.

Thanks!

Answer 1

The answer is no.

Counterexamples for $\kappa = \omega$ are given by sheaves on a compact topological space $X$, where $\Gamma(X,-)$ commutes with filtered colimits. However, the condition that you ask for implies that $\mathcal E = Sh(X)$ is compactly generated, and $Sh(X)$ is rarely compactly generated. If it were, then also its category of opens would be as an $\omega$-accessible localization, but $O(X)$ is of course usually not compactly generated (it often contains no compact beyond $X$ itself, e.g. if $X$ is Hausdorff and connected).

For higher (regular) $\kappa$, one can give similar counterexamples: one simply needs to find a topological space $X$ which is $\kappa$-compact in the obvious sense, but such that $O(X)$ is not $\kappa$-compactly generated; e.g. one can start with an arbitrary $X$ such that $O(X)$ is not $\kappa$-compactly generated, and add a point at infinity to get a compact space with the same "small" opens, namely take $X\cup\{\infty\}$ where the opens are those of $X$, and $X\cup\{\infty\}$ itself. This is clearly a compact space, and if $O(X\cup\{\infty\})$ is $\kappa$-compactly generated, then so is $O(X)$ itself.

There are many examples of spaces with $O(X)$ not $\kappa$-compactly generated, i.e. $X$ does not have a pre-basis of $\kappa$-compact opens (for $\kappa= \omega_1$, an $\omega_1$-compact open is a Lindelöf open, so you'd be looking for a space which does not have a basis of Lindelöf opens). One can find such things, see e.g. the example here .