Classical and t-projective objects

Proof. Suppose first that \(M[0]\) is t-projective, so that \(\hom _{\D (\Aa )}(M[0],-)\colon \D (\Aa ) \to \Sp \) is t-exact. By Corollary 6.3.17, this functor restricts to an exact functor on hearts \(\D (\Aa )^{\heartsuit } \to \Sp ^{\heartsuit }\), i.e., \(\Aa \to \Ab \). This restricted functor is \(\Hom _{\Aa }(M,-)\), showing that \(M\) is projective in \(\Aa \).

Conversely, assume that \(\Hom _{\Aa }(M,-)\colon \Aa \to \Ab \) is exact. Just as in Corollary 6.2.9, this implies that for any chain complex \(C_{\bullet }\) in \(\Aa \) we have an isomorphism \[ \pi _n \hom _{\Kk (\Aa )}(M[0], C_{\bullet }) \cong \Hom _{\Aa }(M,H_n(C)). \] In particular, the functor \(\hom _{\Kk (\Aa )}(M[0],-)\colon \Kk (\Aa ) \to \Sp \) inverts quasi-isomorphisms. The same argument as in Lemma 6.2.10 shows that it is corepresented by \(M[0] \in \D (\Aa )\), giving a natural isomorphism \(\hom _{\D (\Aa )}(M[0],-) \cong \hom _{\Kk (\Aa )}(M[0],-)\). It follows that \(\pi _n\hom _{\D (\Aa )}(M[0],X) \cong \Hom _{\Aa }(M,H_n(X))\) for all complexes \(X \in \D (\Aa )\), showing that \(\hom _{\D (\Aa )}(M[0],X)\) is connective whenever \(X\) is. □