Abstract
We give a general method of constructing positive stable model structures for symmetric spectra over an abstract simplicial symmetric monoidal model category. The method is based on systematic localization, in Hirschhorn’s sense, of a certain positive projective model structure on spectra, where positivity basically means the truncation of the zero level. The localization is by the set of stabilizing morphisms or their truncated version.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Cisinski, D.-C., Déglise, F.: Triangulated categories of mixed motives. Preprint 2013 available at http://www.math.univ-toulouse.fr/~dcisinsk/
Elmendorf, A.D., Kriz, I.,Mandell, M.A., May, J.P.: Rings,Modules, and algebras in stable homotopy theory. With an appendix by M. Cole. Mathematical Surveys and Monographs 47 AMS (1997)
Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Progress in Mathematics 174, Birkhäuser (1999)
Gorchinskiy, S., Guletskiĭ, V.: Symmetric powers in abstract homotopy categories. Adv. Math. 292, 707–754 (2016)
Hirschhorn, P.: Model categories and their localizations. Math. Surveys and Monographs 99. AMS (2003)
Hornbostel, J.: Preorientations of the derived motivic multiplicative group. Alg. Geom. Topology 13, 2667–2712 (2013)
Hovey, M.: Model categories. Math. Surveys and monographs 63 AMS (1999)
Hovey, M.: Spectra and symmetric spectra in general model categories. J. Pure Appl. Algebra 165(1), 63–127 (2001)
Hovey, M., Shipley, B., Smith, J.: Symmetric Spectra. J. AMS 13(1), 149–208 (2000)
Jardine, J.F.: Motivic symmetric spectra. Doc. Math., 445–553 (2000)
Palacios Baldeon, J.: Geometric symmetric powers in the motivic homotopy category. Preprint arXiv:1411.3279v3
Mandell, M.A., May, J.P., Schwede, S., Shipley, B.: Model categories of diagram spectra. Proc. London Math. Soc. (3) 82(2), 441–512 (2001)
Pavlov, D., Scholbach, J.: Symmetric operads in abstract symmetric spectra. Preprint arXiv:1410.5699v1
Schlichtkrull, C.: The homotopy infinite symmetric product represents stable homotopy. Algebr. Geom. Topol. 7, 1963–1977 (2007)
Shipley, B.: A convenient model category for commutative ring spectra. Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic K-theory. Contemporary Mathematics 346, 473–483 AMS (2004)
Vicinsky, D.: The Homotopy Calculus of Categories and Graphs PhD Thesis. https://scholarsbank.uoregon.edu/xmlui/handle/1794/19283?show=full (2015)
Acknowledgments
Open access funding provided by University of Liverpool. The authors are grateful to Peter May, who has drawn our attention to positive model structures in topology, and to Joseph Ayoub for useful comments on homotopy type under the action of finite groups. We are also grateful to the anonymous referee whose comments helped to improve the exposition. The paper is written in the framework of the EPSRC grant EP/I034017/1. The first named author acknowledges the support of the grants MK-5215.2015.1, RFBR 14-01-00178, Dmitry Zimin’s Dynasty Foundation and Laboratory of Mirror Symmetry NRU HSE, RF government grant, ag. No.14.641.31.0001.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Gorchinskiy, S., Guletskiĭ, V. Positive Model Structures for Abstract Symmetric Spectra. Appl Categor Struct 26, 29–46 (2018). https://doi.org/10.1007/s10485-016-9480-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-016-9480-9
Keywords
- Symmetric monoidal model category
- Cofibrantly generated model category
- Localization of a model structure
- Quillen functors
- Symmetric spectra
- Stable model structure
- Stable homotopy category