Corollary 4.3.30 (Standard presentation). In the situation of Lemma 4.3.22, every spectrum \(X \in \Sp (C)\) is naturally isomorphic to the spectrum associated to its underlying prespectrum: \[ \colim _{n \geq 0} \Sigma ^{\infty -n} X_n \quad \xrightarrow {\cong } \quad X, \] where \(X_n=\Omega ^{\infty -n}X\).
Proof. Let \(Y\) be a spectrum in \(\Sp (C)\). Mapping out of the displayed colimit gives \[ \lim _n \Hom _{\Sp (C)}(\Sigma ^{\infty -n}X_n,Y) \simeq \lim _n \Hom _{C_*}(X_n,\Omega ^{\infty -n}Y). \] The transition maps in this limit are precisely the compatibility conditions for a map of prespectra \(X \to Y\). Hence the right-hand side identifies with \(\Hom _{\PSp (C)}(X,Y)\). By the spectrification adjunction from Proposition 4.3.19, this is naturally equivalent to \(\Hom _{\Sp (C)}(X^{\mathrm {sp}},Y)\). The claim follows from Yoneda. □
Generated from the authoritative LaTeX source.