Spectrum objects

Definition 4.3.4 (Spectrum objects). Let \(C\) be an \(\infty \)-category admitting finite limits. We define its \(\infty \)-category of spectrum objects, or stabilization, as the full subcategory \[ \Sp (C) \subseteq \PSp (C) \] spanned by those prespectra \(X\) for which every structure map \(\sigma ^X_n\colon X_n \to \Omega X_{n+1}\) is an isomorphism. We denote the evaluation functors by \[ (-)_n\colon \Sp (C) \to C_*. \] The functor \((-)_0\colon \Sp (C) \to C_*\) will also be denoted by \(\Omega ^{\infty }\); when the target is written as \(C\), we implicitly compose with the forgetful functor \(C_* \to C\).