t-projective objects are connective

Observation 6.4.2. Every t-projective object \(P\) is connective. To see this, consider the truncation \(\tau _{\leq -1}P\). By t-exactness, the mapping spectrum \(\hom _C(P,\tau _{\leq -1} P)\) lies in \(\Sp _{\leq -1}\), and it follows that \[ 0 = \pi _0 \hom _C(P,\tau _{\leq -1} P) \cong \pi _0 \Hom _C(\tau _{\leq -1}P, \tau _{\leq -1}P), \] where the last identification holds by adjunction. In particular, the identity on \(\tau _{\leq -1}P\) is homotopic to the zero map, forcing \(\tau _{\leq -1}P = 0\), hence \(P \in C_{\geq 0}\).