Given a site $C$, there are various standard notions for an object $X \in C$ being compact. For instance:
Every covering family $\lbrace U_i \to X \rbrace$ has a finite subfamily that is still covering.
The functor $C(X,-)$ commutes with filtered colimits.
After Yoneda-embedding, the functor $Sh_C(X, -)$ commutes with filtered colimits.
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?