Sphere spectrum

Definition 4.4.2 (Sphere spectrum). We define the sphere spectrum as \(\S := \Sigma ^{\infty }(S^0) \simeq \Sigma ^{\infty }_+(*) \in \Sp \).

For an anima \(X\), we will frequently use the alternative notation \[ \S [X] \quad := \quad \Sigma ^{\infty }_+(X) \] for the unreduced suspension spectrum. We wish to think of the spectrum \(\S [X]\) as the ‘free spectrum generated by \(X\)’. This notation mimics the notation for the free abelian group \(\Z [S]\) generated by a set \(S\), characterized by the property that group homomorphisms \(\Z [S] \to A\) correspond to maps of sets \(S \to A\).