×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] [AT] Araki, S., Toda, H.: Multiplicative structures for modq cohomology theories I. Osaka J. Math.2, 71-115 (1965) · Zbl 0129.15201
[2] [AT] Araki, S., Toda, H.: Multiplicative structures for modq cohomology theories II. Osaka J. Math.3, 81-120 (1966) · Zbl 0152.22102
[3] [BBD] Beilinson, A.A., Bernstein, J., Deligne, P.: Analyse et topologie sur les ?spaces singuliers. Ast?risque 100, Soc. Math. France 1982
[4] [BC] Brown, E.H., Comenetz, M.: Pontrjagin duality for generalized homology and cohomology theories. Am. J. Math.98, 1-27 (1976) · Zbl 0325.55008 · doi:10.2307/2373610
[5] [C] Cohen, J.M.: The decomposition of stable homotopy. Ann. Math.87, 305-320 (1968) · Zbl 0162.55102 · doi:10.2307/1970586
[6] [H] Heller, A.: Extraordinary homology and chain complexes., In: Eilenberg, S., Harrison, D.K., MacLane, S., R?hrl, H. (eds.) Proceddings of the Conference on Categorical Algebra, La Jolla (1965) 355-365. Berlin Heidelberg New York: Springer 1966
[7] [HS] Hinich, V.A., Schechtman, V.V.: Geometry of a category of complexes and algebraicK-theory. Duke Math. J.52, 399-430 (1985) · Zbl 0574.55020 · doi:10.1215/S0012-7094-85-05220-2
[8] [L] Landsburg S.E.:K-theory and patching for categories of complexes. Duke Math. J.62, 359-384 (1991) · Zbl 0747.18012 · doi:10.1215/S0012-7094-91-06214-9
[9] [N1] Meeman, A.: Some new axioms for triangulated categories (formerly: Triangulated categories are all wrong). J. Algebra (to appear)
[10] [N2] Neeman, A.: The Brown Representability Theorem and phantomless triangulated categories. (Preprint) · Zbl 0799.18006
[11] [N3] Neeman, A.:K-theory for triangulated categories II: homological functors. (Preprint)
[12] [TT] Thomason, R. Trobaugh, T.: Higher algebraicK-theory of schemes and of derived categories. In: The Grothendieck Festschrift, (a collection of articles to mark Grothendieck’s 60th birthday) Volume 3, pp. 247-435. Basel: Birkh?user 1990
[13] [Wa] Waldhausen, F.: AlgebraicK-theory of spaces. Lec. Notes in Math. vol. 1126, pp. 318-419, 1985 · doi:10.1007/BFb0074449
[14] [Wh] Whitehead, G.W.: Recent advances in homotopy theory. Am. Math. Soc. regional conference series in Math., Vol. 5, 1970 · Zbl 0217.48601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.