Prespectra

Definition 4.3.3 (Prespectra). Let \(C\) be an \(\infty \)-category admitting finite limits. Write \[ C_*^{\N } := \prod _{n \geq 0} C_* \] for the \(\infty \)-category of sequences of pointed objects of \(C\). We define the shift functor and the levelwise loop functor by \[ \sh \colon C_*^{\N } \to C_*^{\N }, \qquad (X_n)_{n \geq 0} \mapsto (X_{n+1})_{n \geq 0}, \] and \[ \Omega \colon C_*^{\N } \to C_*^{\N }, \qquad (X_n)_{n \geq 0} \mapsto (\Omega X_n)_{n \geq 0}. \] The \(\infty \)-category of prespectra in \(C\) is defined as \[ \PSp (C) := C_*^{\N } \times _{C_*^{\N } \times C_*^{\N }} \Ar (C_*^{\N }), \] where the map \(C_*^{\N } \to C_*^{\N } \times C_*^{\N }\) is \((\id ,\Omega \sh )\) and the map \(\Ar (C_*^{\N }) \to C_*^{\N } \times C_*^{\N }\) is the source-target functor \((s,t)\).