# Compact objects in triangulated and infinity categories

Hello,

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 2.4.1.2, 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

• Why do you think that a homotopy colimit is not given by a filtered limit in the category of complexes?:) – Mikhail Bondarko Oct 20 '12 at 9:30
• I meant that I don't see it personally. Could you hint how can I see it? – Sasha Oct 20 '12 at 10:14
• I believe that (2) of Proposition 1.4.4.1 of math.harvard.edu/~lurie/papers/HigherAlgebra.pdf shows that the two notions of compactness are equivalent in a stable $\infty$-category. – Akhil Mathew Oct 20 '12 at 13:02
• As for (2), it's very often the case that filtered colimits in model categories preserve equivalences and hence compute the correct homotopy colimits. This is the case for example in any "compactly generated model category", cf. Prop. 2.2 in arxiv.org/pdf/math/0503269v5.pdf. It might be this applies in your case. – Marc Hoyois Oct 20 '12 at 18:24
• @Akhil and @Marc - Thank you, I will think about your comments. – Sasha Oct 20 '12 at 18:52