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Stable homotopy as a triangulated functor. (English) Zbl 0793.55007
Let $${\mathcal S}$$ denote the stable category of spectra and $${\mathcal D} \mathbb{Z} [\frac 12]$$ the derived category of $$Z[\frac 12]$$-modules. The author constructs a triangulated functor $$\Pi:{\mathcal S} \to {\mathcal D} \mathbb{Z} [\frac 12]$$ lifting the stable homotopy group functor with 2 inverted. As a triangulated functor having this property $$\Pi$$ is unique up to natural isomorphism. It is constructed as a right adjoint of the Moore spectrum functor $${\mathcal D} \mathbb{Z} [\frac 12] \to {\mathcal S} [\frac 12]$$. Functors of this kind have previously been considered by A. Heller [Proc. Conf. Categor. Algebra, La Jolla 1965, 355-365 (1966; Zbl 0188.284)].
The functor $$\Pi$$ is compatible with the smash and tensor product: there is a natural transformation $$\mu_{X,Y}:\Pi (X) \otimes \Pi (Y) \to \Pi (X \wedge Y)$$. $$\Pi$$ cannot be a monoidal functor (the Moore spectrum of $$\mathbb{Z}/3$$ provides a counterexample), but if 6 is inverted the diagram $\begin{tikzcd} [column sep = 2cm]\Pi (X) \otimes \Pi (Y) \otimes \Pi (Z) \ar[r,"\mu_{X,Y} \otimes id"]\ar[d,"id \otimes \mu_{Y,Z}"'] & \pi (X \wedge Y) \otimes \Pi (Z) \ar[d,"\mu_{X \wedge Y,Z}"]\\ \Pi (X) \otimes \Pi (Y \wedge Z)) \ar[r,"\mu_{X,Y \wedge Z}"'] & \Pi (X \wedge Y \wedge Z) \end{tikzcd}$ commutes.

##### MSC:
 55P42 Stable homotopy theory, spectra
##### Keywords:
derived category; triangulated functor; Moore spectrum
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##### References:
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