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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?

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Some of these general questions are treated in Exposé VI of SGA 4. –  Martin Brandenburg Apr 30 '12 at 8:55
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