Proposition 6.3.21 (Heart of the Postnikov t-structure). The functor \(\bB ^{\infty }\colon \Ab \hookrightarrow \Sp \) induces an equivalence \[ \bB ^{\infty }\colon \Ab \iso \Sp ^{\heartsuit }. \]
Proof. By the recognition principle for connective spectra (Theorem 5.4.6), the functor \(\bB ^{\infty }\) induces an equivalence \(\bB ^{\infty }\colon \CGrp (\An ) \iso \Sp _{\geq 0}\). By Corollary 6.3.19, we may identify \(\Sp ^{\heartsuit }\) with the subcategory of \(\CGrp (\An )\) spanned by the 0-truncated objects. Note that object in \(\CGrp (\An )\) is \(0\)-truncated if and only if its underlying anima is 0-truncated, which in turn happens if and only if it is contained in the full subcategory \(\Set \subseteq \An \). All in all, we get \(\Ab = \CGrp (\Set ) \iso \CGrp (\An )_{\leq 0} \iso \Sp ^{\heartsuit }\). □
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