Lemma 16.5.8. If \(C\) is stable, the evaluation functor \(\Omega ^\infty \colon \Sp ^{\exc }(C) \to C\) is an equivalence.
Proof. When \(C\) is stable, a reduced functor \(\An _*^{\fin }\to C\) is excisive if and only if it preserves finite colimits. Hence \[ \Sp ^{\exc }(C) = \Exc _*(\An _*^{\fin },C) \simeq \Fun _*^{\mathrm {rex}}(\An _*^{\fin },C). \] Evaluation at \(S^0\) is an equivalence by the universal property of \(\An _*^{\fin }\) from Notation 16.5.2. □
Generated from the authoritative LaTeX source.