Applied Categorical Structures

, Volume 23, Issue 3, pp 487–505 | Cite as

Homological Localisation of Model Categories

  • David BarnesEmail author
  • Constanze Roitzheim


One of the most useful methods for studying the stable homotopy category is localising at some spectrum E. For an arbitrary stable model category we introduce a candidate for the E–localisation of this model category. We study the properties of this new construction and relate it to some well–known categories.


Stable model categories Bousfield localisation 

Mathematics Subject Classifications (2010)

55P42 55P60 18E30 16D90 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bergner, J.E.: Homotopy fiber products of homotopy theories. Israel J. Math. 185, 389–411 (2011)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Bousfield, A.K.: The localization of spectra with respect to homology. Topology 18(4), 257–281 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Barnes, D., Roitzheim, C.: Local framings. N. Y. J. Math. 17, 513–552 (2011)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Barnes, D., Roitzheim, C.: Stable left and right bousfield localisations. Glasg. Math. J. FirstView, 1–30 (2013)Google Scholar
  5. 5.
    Dugger, D., Shipley, B.: Enriched model categories and an application to additive endomorphism spectra. Theor. Appl. Categ. 18, 400–439 (2007). ElectroniczbMATHMathSciNetGoogle Scholar
  6. 6.
    Elmendorf, A.D., Kriz, I., Mandell, M.A., May, J.P.: Rings, modules, and algebras in stable homotopy theory, volume 47 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (1997). With an appendix by M. ColeGoogle Scholar
  7. 7.
    Goerss, P.G., Jardine, J.F.: Simplicial homotopy theory, volume 174 of Progress in Mathematics. Basel, Birkhäuser Verlag (1999)CrossRefGoogle Scholar
  8. 8.
    Gutiérrez, J.J.: Homological localizations of Eilenberg–Mac Lane spectra. Forum Math. 22(2), 349–356 (2010)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Hirschhorn, P.S.: Model categories and their localizations, volume 99 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2003)Google Scholar
  10. 10.
    Hovey, M.: Model categories, volume 63 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (1999)Google Scholar
  11. 11.
    Hovey, M.: Spectra and symmetric spectra in general model categories. J. Pure Appl. Algebra 165(1), 63–127 (2001)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Hovey, M., Strickland, N.P.: Morava K-theories and localisation. Mem. Am. Math. Soc. 139(666), viii+100 (1999)MathSciNetGoogle Scholar
  13. 13.
    Hovey, M., Shipley, B., Smith, J.: Symmetric spectra. J. Am. Math. Soc. 13(1), 149–208 (2000)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Lenhardt, F.: Stable frames in model categories. J. Pure Appl. Algebra 216(5), 1080–1091 (2012)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Ravenel, D.C.: Nilpotence and periodicity in stable homotopy theory, volume 128 of Annals of Mathematics Studies. Princeton University Press, Princeton (1992). Appendix C by Jeff SmithGoogle Scholar
  16. 16.
    Roitzheim, C.: Rigidity and exotic models for the K-local stable homotopy category. Geom. Topol. 11, 1855–1886 (2007)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Schwede, S.: The stable homotopy category is rigid. Ann. Math. (2) 166(3), 837–863 (2007)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Schwede, S., Shipley, B.: Algebras and modules in monoidal model categories. Proc. London Math. Soc. (3) 80(2), 491–511 (2000)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Schwede, S., Shipley, B.: Stable model categories are categories of modules. Topology 42(1), 103–153 (2003)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Pure Mathematics Research Centre, School of Mathematics and PhysicsQueen’s University BelfastBelfastUK
  2. 2.School of Mathematics, Statistics and Actuarial ScienceUniversity of KentCanterburyUK

Personalised recommendations