Spectrum of spectrum in the stable homotopy category

89 Views Asked by At

Let $\mathcal{E}=(E_0,E_1,\cdots)$ be an $S^1$-spectrum. Define $\Sigma \mathcal{E}$ to be the spectrum with $(\Sigma \mathcal{E})_n=E_{n+1}$. Then, consider the spectrum $\tilde{\mathcal{E}} =(\mathcal{E}, \Sigma\mathcal{E}, \Sigma^2 \mathcal{E},\cdots)$. The bonding maps are maps of spectra induced by those of $\mathcal{E}$, $S^1\wedge E_n\to E_{n+1}$.

Can $\tilde{\mathcal{E}}$ be considered as an object in $SH$, the stable homotopy category of $S^1$-spectra? If yes, is there an equivalence $\mathcal{E}\simeq \tilde{\mathcal{E}} $ in $SH$?