Compact generation criterion

Proposition 22.3.4 (Compact generation criterion, cf. Hovey et al. (1997), Corollary 2.3.12). Let \(C\) be a stable \(\infty \)-category with small colimits and let \(\Gg \) be a small collection of compact objects. The following conditions are equivalent:

(1)

The collection \(\Gg \) generates \(C\).

(2)

The functors \[ \Hom _C(G[n],-)\colon C\to \An , \qquad G\in \Gg ,\ n\in \Z , \] are jointly conservative.

(3)

Every object of \(C\) is a filtered colimit of objects in the smallest thick subcategory of \(C\) containing \(\Gg \).

If these conditions hold, then \(C\) is compactly generated and its compact objects form the smallest thick subcategory of \(C\) containing \(\Gg \).