Proposition 6.3.4 (Postnikov t-structure on spectra). Define \(\Sp _{\geq 0} \subseteq \Sp \) to be the full subcategory of connective spectra, i.e. those \(X\) with \(\pi _k(X) = 0\) for \(k < 0\), and \(\Sp _{\leq 0} \subseteq \Sp \) the full subcategory of coconnective spectra, i.e. those with \(\pi _k(X) = 0\) for \(k > 0\). Then the pair \((\Sp _{\geq 0}, \Sp _{\leq 0})\) defines a t-structure on \(\Sp \), called the Postnikov t-structure.
Proof. Property (1) is immediate from the behavior of homotopy groups under shifts: \(\pi _k(X[1]) \cong \pi _{k-1}(X)\).
For property (2), let \(X \in \Sp _{\geq 0}\) and \(Y \in \Sp _{\leq 0}\). By the recognition principle for connective spectra (Theorem 5.4.6), we may write \(X \simeq \bB ^{\infty }(A)\) for some \(A \in \CGrp (\An )\). By adjunction, \[ \Hom _{\Sp }(X, Y[-1]) \quad \cong \quad \Hom _{\CGrp (\An )}(A, \Omega ^{\infty }(Y[-1])). \] But \(\Omega ^{\infty }(Y[-1])\) is contractible: its homotopy groups \(\pi _k(\Omega ^{\infty }(Y[-1])) \cong \pi _k(Y[-1]) \cong \pi _{k+1}(Y)\) vanish for all \(k \geq 0\) since \(Y \in \Sp _{\leq 0}\).
For property (3), given a spectrum \(X\), we define \(\tau _{\geq 0} X := \bB ^{\infty }\Omega ^{\infty }X\). The counit of the adjunction \(\bB ^{\infty } \dashv \Omega ^{\infty }\) provides a map \(\tau _{\geq 0} X \to X\). Define \(\tau _{\leq -1} X\) as its cofiber. From the long exact sequence on homotopy groups (Proposition 4.4.30), one verifies that \(\tau _{\geq 0} X\) is connective and \(\tau _{\leq -1} X\) has homotopy groups concentrated in degrees \(\leq -1\). □
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