I think of a compact object in a category as an object, $Hom$ from which commutes with filtered colimits.
I guess that in an infinity category, one also defines a compact object as an object, $Hom$ from which commutes with homotopy filtered colimits.
1) In a derived category (in the classical sense), one defines a compact object as an object, $Hom$ from which commutes with direct sums. Is this equivalent to this object being compact in the above sense, when one thinks of it now in the derived category, as an infinity category?
2) In the paper of Thomason-Trobaugh, formula 188.8.131.52, it is written that $Hom$ from a perfect complex, commutes with filtered colimits; Here $Hom$ is in the derived category, but the colimit is in the category of complexes. How should I relate it to the notion of compactness? I mean, the colimit is not a homotopy colimit in this formulation.
Thank you, Sasha