Compact space in site -> compact object in topos

Question

Given a site $C$, there are various standard notions for an object $X \in C$ being compact. For instance:

1. Every covering family $\lbrace U_i \to X \rbrace$ has a finite subfamily that is still covering.

2. The functor $C(X,-)$ commutes with filtered colimits.

3. After Yoneda-embedding, the functor $Sh_C(X, -)$ commutes with filtered colimits.

4. After $\infty$-Yoneda-embedding, the functor $\infty Sh_C(X, -)$ commutes with filtered $\infty$-colimits.

These notions are closely related but subtly different. For instance for $C = Top$ it is well known that the first two are not equivalent without further fine-tuning.

What can one say about the relation of 1. to 3. and 4. ?

It seems to me that one can say for instance: compactness in the first sense implies that $Sh_C(X,-)$ commutes with mono-filtered colimits, and this should generalize to the $\infty$-case in the suitable sense.

What else can one say?

Answer 1

The question that you ask might be better phrased intrinsically without referring to sites. Fix an $0 \leq n \leq \infty$ and let $\mathbf{X}$ be an $n$-topos. Recall that an object $X \in \mathbf{X}$ is quasi-compact if for every effective epimorphism of the form $\coprod_{i \in I} U_i \to X$ there is a finite subset $I_0 \subseteq I$ such that $\coprod_{i \in I_0} U_i \to X$ is an effective epimorphism. On the other hand, let us say that $X \in \mathbf{X}$ is small if the functor $Map_{\mathbf{X}}(X,-)$ preserves filtered colimits. Your question can then be phrased as follows:

What is the relation of between the property of being quasi-compact and the property of being small?

Using the fact that colimits in an $n$-topos are universal one can easily verify that $X \in \mathbf{X}$ is quasi-compact or small if and only if the terminal object in $\mathbf{X}_{/X}$ is quasi-compact or small respectively. We can hence assume that $X$ itself was the terminal object $\ast \in \mathbf{X}$. Now it is not hard to prove that $\ast \in \mathbf{X}$ is quasi-compact if and only if $\ast$ is small when regarded as an object of the underlying $0$-topos of $\mathbf{X}$ (i.e. the full subcategory spanned by $(-1)$-truncated objects). Hence we see that $\ast$ being small implies $\ast$ being quasi-compact. However, when $n > 0$ the inverse implication is false in general.

Here is a counterexample that works for every $0 < n \leq \infty$. Let $G$ be a free group on an infinite set of generators and let $\mathbf{X}$ be the $n$-topos of $(n-1)$-types equipped with a $G$-action. Then the terminal object $\ast$ is quasi-compact but the functor represented by $\ast$ is the homotopy fixed point functor, which does not commute with filtered colimits for such $G$.