Basic compact generators

Example 22.3.10. The categories \(\An \), \(\An _*\), and \(\Sp \) are compactly generated. For animae and pointed animae this follows from Lemma 22.3.9. The sphere spectrum is a compact generator of \(\Sp \): compactness follows because \(\Hom _{\Sp }(\S ,-)\simeq \Omega ^\infty \) preserves filtered colimits by Lemma 4.3.21, while its shifts jointly detect isomorphisms by Corollary 4.4.27; now apply Proposition 22.3.4.

For every associative ring spectrum \(A\), the free module \(A\) is a compact generator of \(\LMod _A\) by Lemma 8.1.8; for every commutative ring \(R\), the complex \(R[0]\) is a compact generator of \(\D (R)\) by Corollary 20.2.5. Consequently, \[ \Sp \simeq \Ind (\Sp ^\omega ), \qquad \LMod _A\simeq \Ind (\Perf (A)), \qquad \D (R)\simeq \Ind (\Perf (R)). \] The Eilenberg–MacLane comparison of Corollary 8.3.4 identifies the last presentation with the corresponding presentation of \(\Mod _{HR}\).