Spectrum

Definition 3.2.1 (Spectrum). A spectrum \(E\) is a sequence of pointed animae \((E_m)_{m \geq 0}\) together with isomorphisms \[ \sigma _m\colon E_m \iso \Omega E_{m+1} \] for all \(m \geq 0\). We call the maps \(\sigma _m\) the structure maps of \(E\).

For convenience, we extend the sequence to all integer indices by setting \[ E_k := \Omega ^{-k}E_0 \qquad (k<0). \] With this convention, the structure maps extend to all \(k\in \Z \): for \(k<0\), we take \(\sigma _k\) to be the identity map under the identification \[ E_k=\Omega ^{-k}E_0=\Omega (\Omega ^{-k-1}E_0)=\Omega E_{k+1}. \]