Suspension is the shift on spectrum objects

Proof. By Lemma 4.3.8, finite limits in \(\PSp (C)\) are computed pointwise. Since \(\Omega \) preserves finite limits, the full subcategory \(\Sp (C) \subseteq \PSp (C)\) is closed under finite limits. It is also pointed: the constant zero prespectrum is both initial and terminal, since the zero object of \(C_*\) is both initial and terminal.

It remains, by Theorem 4.2.2, to show that the loop functor on \(\Sp (C)\) is an equivalence. Define the shift functor \[ \shift \colon \Sp (C) \to \Sp (C), \qquad \shift (X)_n:=X_{n+1}, \] with structure maps inherited from \(X\). This functor is an equivalence: an inverse sends \(X\) to the spectrum \(\shift ^{-1}X\) with \[ (\shift ^{-1}X)_0:=\Omega X_0 \qquadtext { and } \qquad (\shift ^{-1}X)_n:=X_{n-1}\quad (n\geq 1), \] whose structure map in degree \(0\) is the identity of \(\Omega X_0\), and whose structure map in degree \(n\geq 1\) is \(\sigma ^X_{n-1}\).

We claim that there is a natural isomorphism \[ \alpha \colon X \xrightarrow {\cong } \Omega (\shift X). \] By Lemma 4.3.9, it suffices to construct this map after passing to the even-level model. There we set \[ \alpha _{2k}:=\sigma ^X_{2k}\colon X_{2k} \xrightarrow {\cong } \Omega X_{2k+1} = \Omega (\shift X)_{2k}. \] We must check compatibility with the two-step structure maps. For \(X\), the relevant structure map is \[ \widetilde {\sigma }^X_k = \left (X_{2k}\xrightarrow {\sigma ^X_{2k}}\Omega X_{2k+1} \xrightarrow {\Omega (\sigma ^X_{2k+1})}\Omega ^2X_{2k+2}\right ). \] For \(\Omega (\shift X)\), finite limits are computed pointwise, but its two-step structure map is not obtained by merely writing down the same composite with an extra \(\Omega \) in front. After precomposing with \(\alpha _{2k}=\sigma ^X_{2k}\), the first composite in the compatibility square is \[ X_{2k} \xrightarrow {\sigma ^X_{2k}} \Omega X_{2k+1} \xrightarrow {\Omega (\sigma ^X_{2k+1})} \Omega ^2X_{2k+2} \xrightarrow {\Omega (\Omega (\sigma ^X_{2k+2}))} \Omega (\Omega ^2X_{2k+3}) \xrightarrow {\cong } \Omega ^2(\Omega X_{2k+3}), \] where the final arrow is the canonical comparison coming from the fact that \(\Omega ^2\) preserves loop objects. The other composite in the compatibility square is \[ X_{2k} \xrightarrow {\sigma ^X_{2k}} \Omega X_{2k+1} \xrightarrow {\Omega (\sigma ^X_{2k+1})} \Omega ^2X_{2k+2} \xrightarrow {\Omega ^2(\sigma ^X_{2k+2})} \Omega ^2(\Omega X_{2k+3}). \] Thus the two composites whose equality is required differ only by the resulting automorphism of \(\Omega ^3X_{2k+3}\). This automorphism is the identity by Proposition 4.3.13. Hence \(\alpha \) is an isomorphism of spectra.

Consequently \(\id _{\Sp (C)}\simeq \Omega \circ \shift \). Since \(\shift \) is an equivalence, \(\Omega \) is an equivalence as well. □