Stable ∞-categories have no nonzero projective objects

Warning 6.4.20. It is not reasonable to ask for \(P\) to be projective in the stable \(\infty \)-category \(C\) itself, since the only projective objects in \(C\) are the zero objects. To see this, assume that \(P\) is a projective object of \(C\). By Lemma 6.4.18, the shift \(N[1] \cong 0 \sqcup _N 0\) of any object \(N \in C\) is the geometric realization of the simplicial object with \(N^{\oplus n}\) in degree \(n\). We then obtain equivalences \[ \Hom _{C}(P[-1],N) \simeq \Hom _{C}(P,N[1]) \simeq \abs { [n] \mapsto \Hom _{C}(P,N^{\oplus n}) }. \] Since the right-hand side is connected, we see that \(\pi _0\Hom _{C}(P[-1],N) = 0\). Replacing \(N\) by \(N[-k]\) for all \(k \in \N \) then gives that \(\pi _k\Hom _{C}(P[-1],N) = 0\) for all \(k\), hence \(\Hom _{C}(P[-1],N) = 0\), showing that \(P\) is the zero object.