Remark 1.1. The notion of compactly generated categories is a standard one in homotopy theory, especially in conjunction with themes such as Brown representability and Bousfield localization. Schwede and Shipley [SSh] prove in great generality that compactly generated categories can be expressed as categories of modules.
In algebraic geometry, the importance of the interplay between compact and perfect objects was originally recognized and put to great use by Thomason [TT]. These ideas were combined with homotopical techniques by Bökstedt and Neeman [BoN], and further developed and enhanced by Neeman [N1, N2] and many others [Ke, BV, To1]. The key property of derived categories of quasi-coherent sheaves on quasi-compact, separated schemes identified in these papers is that on the one hand, they are compactly generated, and on the other hand, their compact and perfect objects coincide. Such categories appear as unital algebraic stable homotopy categories in the general axiomatic framework developed by Hovey, Palmieri and Strickland [HPS]. This combination of properties underlies the definition of a perfect stack.