Skip to main content
SpringerLink
Log in

Rigidification of Algebras Over Essentially Algebraic Theories

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

Badzioch and Bergner proved a rigidification theorem saying that each homotopy simplicial algebra is weakly equivalent to a simplicial algebra. The question is whether this result can be extended from algebraic theories to finite limit theories and from simplicial sets to more general monoidal model categories. We will present some answers to this question.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adámek, J., Rosický, J.: Locally Presentable and Accessible Categories. Cambridge University Press (1994)

  2. Badzioch, B.: Algebraic theories in homotopy theory. Ann. Math. 155, 895–913 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  3. Barwick, C.: On left and right model categories and left and right Bousfield localizations. Homology, Homotopy Appl. 12, 245–320 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bergner, J.E.: Rigidification of algebras over multi-sorted theories. Alg. Geom. Topology 6, 1925–1955 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bergner, J.E.: Simplicial monoids and Segal categories. Contemp. Math. 431, 59–83 (2007)

    Article  MathSciNet  Google Scholar 

  6. Bourke, J.: A colimit decomposition for homotopy algebras in CAT. arXiv:1206.1203

  7. Dundas, B.I., Röndings, O., Østvaer, P.A.: Enriched functors and stable homotopy theory. Doc. Math. 8, 409–488 (2003)

    MATH  MathSciNet  Google Scholar 

  8. Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Birkhäuser (1999)

  9. Hirschhorn, P.S.: Model categories and their localizations. Math. Surveys and Monographs 99, 457 pp. (2003)

    MathSciNet  Google Scholar 

  10. Hirschowitz, A., Simpson, C.: Descent pour les n-champs. arXiv:math.AG/9807049

  11. Hovey, M.: Model categories. Math. Surveys and Monographs 63, 209 pp. (1999)

    MathSciNet  Google Scholar 

  12. Johnstone, P.T.: Sketches of an Elefant. A Topos Theory Compendium, vol. 1. Oxford University Press (2002)

  13. Kelly, G.M.: Structure defined by finite limits in the enriched context, I. Cah. Top. Géom. Diff. XXIII, 3–42 (1982)

    Google Scholar 

  14. Kelly, G.M.: Basic Concepts of Enriched Category Theory. Cambridge University Press (1982)

  15. Kelly, G.M., Schmitt, V.: Notes on enriched categories with colimits of some class. Theor. Appl. Categ. 14, 399–423 (2005)

    MATH  MathSciNet  Google Scholar 

  16. Lack, S., Rosický, J.: Homotopy locally presentable enriched categories (2012, in press)

  17. Lurie, J.: Higher Topos Theory. Princeton University Press (2009)

  18. Mac Lane, S.: Categories for the Working Mathematician. Springer (1998)

  19. Makkai, M., Rosický, J., Vokřínek, L.: On a fat small object argument. arXiv:1304.6974

  20. Power, J.: Enriched Lawvere theories. Theor. Appl. Categ. 6, 83–93 (1999)

    MATH  MathSciNet  Google Scholar 

  21. Rosický, J.: On combinatorial model categories. Appl. Categor. Struct. 17, 303–316 (2009)

    Article  MATH  Google Scholar 

  22. Schwede, S.: Stable homotopical algebra and Γ-spaces. Math. Proc. Camb. Philos. Soc. 126, 329–356 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  23. Simpson, C.: A closed model structure for n-categories, internal Hom, n-stacks and generalized Seifert–Van Kampen theorem. arXiv:alg-geom/9704006

  24. Shulman, M.: Homotopy limits and colimits and enriched homotopy theory. arXiv:math.AT/0610194

  25. Vokřínek, L.: Homotopy weighted colimits. arXiv:1201.2970

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Masaryk University, Faculty of Sciences, Kotlářská 2, 611 37, Brno, Czech Republic

    J. Rosický

Authors
  1. J. Rosický
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Rosický.

Additional information

Supported by the Grant Agency of the Czech republic under grant 201/11/0528.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rosický, J. Rigidification of Algebras Over Essentially Algebraic Theories. Appl Categor Struct 23, 159–175 (2015). https://doi.org/10.1007/s10485-013-9329-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-013-9329-4

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation