Skip to main content

Advertisement

Log in

Homological Localisation of Model Categories

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Bergner, J.E.: Homotopy fiber products of homotopy theories. Israel J. Math. 185, 389–411 (2011)

    MATH  MathSciNet  Google Scholar 

  2. Bousfield, A.K.: The localization of spectra with respect to homology. Topology 18(4), 257–281 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  3. Barnes, D., Roitzheim, C.: Local framings. N. Y. J. Math. 17, 513–552 (2011)

    MATH  MathSciNet  Google Scholar 

  4. Barnes, D., Roitzheim, C.: Stable left and right bousfield localisations. Glasg. Math. J. FirstView, 1–30 (2013)

    Google Scholar 

  5. Dugger, D., Shipley, B.: Enriched model categories and an application to additive endomorphism spectra. Theor. Appl. Categ. 18, 400–439 (2007). Electronic

    MATH  MathSciNet  Google Scholar 

  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. Cole

    Google Scholar 

  7. Goerss, P.G., Jardine, J.F.: Simplicial homotopy theory, volume 174 of Progress in Mathematics. Basel, Birkhäuser Verlag (1999)

    Book  Google Scholar 

  8. Gutiérrez, J.J.: Homological localizations of Eilenberg–Mac Lane spectra. Forum Math. 22(2), 349–356 (2010)

    MATH  MathSciNet  Google Scholar 

  9. Hirschhorn, P.S.: Model categories and their localizations, volume 99 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2003)

    Google Scholar 

  10. Hovey, M.: Model categories, volume 63 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (1999)

    Google Scholar 

  11. Hovey, M.: Spectra and symmetric spectra in general model categories. J. Pure Appl. Algebra 165(1), 63–127 (2001)

    MATH  MathSciNet  Google Scholar 

  12. Hovey, M., Strickland, N.P.: Morava K-theories and localisation. Mem. Am. Math. Soc. 139(666), viii+100 (1999)

    MathSciNet  Google Scholar 

  13. Hovey, M., Shipley, B., Smith, J.: Symmetric spectra. J. Am. Math. Soc. 13(1), 149–208 (2000)

    MATH  MathSciNet  Google Scholar 

  14. Lenhardt, F.: Stable frames in model categories. J. Pure Appl. Algebra 216(5), 1080–1091 (2012)

    MATH  MathSciNet  Google Scholar 

  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 Smith

    Google Scholar 

  16. Roitzheim, C.: Rigidity and exotic models for the K-local stable homotopy category. Geom. Topol. 11, 1855–1886 (2007)

    MATH  MathSciNet  Google Scholar 

  17. Schwede, S.: The stable homotopy category is rigid. Ann. Math. (2) 166(3), 837–863 (2007)

    MATH  MathSciNet  Google Scholar 

  18. Schwede, S., Shipley, B.: Algebras and modules in monoidal model categories. Proc. London Math. Soc. (3) 80(2), 491–511 (2000)

    MATH  MathSciNet  Google Scholar 

  19. Schwede, S., Shipley, B.: Stable model categories are categories of modules. Topology 42(1), 103–153 (2003)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to David Barnes.

Additional information

The first author was supported by EPSRC grant EP/H026681/1, the second author by EPSRC grant EP/G051348/1.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Barnes, D., Roitzheim, C. Homological Localisation of Model Categories. Appl Categor Struct 23, 487–505 (2015). https://doi.org/10.1007/s10485-013-9340-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-013-9340-9

Keywords

Mathematics Subject Classifications (2010)

Navigation