Suppose $E$ is a connective spectrum, then there exists a natural map in the stable homotopy category $\mathcal{SHC}$, $E \rightarrow P_0 E$, called the $0$-th Postnikov truncation, which is characterized by the property of being an isomorphism in $\pi_0$ and killing all higher homotopy groups. See for example Schwede, chapter II, 8.

The zeroth-Postnikov truncation is (weakly equivalent to) an Eilenberg-MacLane spectrum $H \pi_0 E$. In either the category of symmetric or orthogonal spectra we do have an explicit model for $HA$ for an abelian group $A$ as $\tilde{A}( \mathbb{S})$ where $\tilde A$ is the reduced linearization functor.

The Postnikov truncation of the sphere spectrum for example is given by the Hurewicz map, which is a natural map of symmetric (or orthogonal) spectra $\mathbb S \rightarrow H \mathbb Z$, defined by sending $x \mapsto 1 \cdot x$.

Can we give in a similar manner for an arbitrary symmetric spectrum a natural map of symmetric spectra $E \rightarrow H \pi_0 E$ which realizes the $0$-th Postnikov truncation in the stable homotopy category? Can we do so with sufficient restrictions on $E$ (e.g. being cofibrant) or with different models of $H \pi_0 E$? (Or alternatively in orthogonal spectra)

  • $\begingroup$ Doesn't remark II.8.2 of Schwede's book suggest that the answer is yes? $\endgroup$ – Drew Heard Apr 5 '18 at 19:33
  • $\begingroup$ Isn't it just a left Bousfield localization? You can use Bousfield's formulas for $P_0E$ and for the map $E\to P_0E$ (which is just fibrant replacement in the local model structure), but are those good enough for what you need? $\endgroup$ – David White Apr 10 '18 at 21:50

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