Compactly generated category
Introduction Compactly generated categories are the tame end of the presentable spectrum. Every object is a filtered colimit of compact ones, and most homological invariants — K-theory, Hochschild homology, cyclic homology — are determined by the small subcategory of compact objects. The idea: a “big” category is recovered, up to filtered colimits, from a small skeleton of finite/perfect/dualizable atoms. Examples are everywhere in algebra and topology: spectra (compact = finite spectra), modules over a ring spectrum (compact = perfect), the derived category of a ring (compact = perfect complexes). The fact that all of homotopy theory’s “large” categories of interest are compactly generated is why the theory of K-theory, traces and dimensions is as workable as it is. ...