Applied Categorical Structures

, Volume 10, Issue 3, pp 195–220 | Cite as

Model Categories in Algebraic Topology

  • Kathryn Hess


This survey of model categories and their applications in algebraic topology is intended as an introduction for non homotopy theorists, in particular category theorists and categorical topologists. We begin by defining model categories and the homotopy-like equivalence relation on their morphisms. We then explore the question of compatibility between monoidal and model structures on a category. We conclude with a presentation of the Sullivan minimal model of rational homotopy theory, including its application to the study of Lusternik–Schnirelmann category.

algebraic homotopy theory Lusternik–Schnirelmann category model category monoidal category rational homotopy theory Sullivan model 


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  1. 1.
    Baues, H.: Algebraic Homotopy, Cambridge Studies in Advanced Mathematics, Vol. 15, Cambridge Univ. Press, 1989.Google Scholar
  2. 2.
    Bousfield, A. K. and Gugenheim, V. K. A.M.: On PL De Rham Theory and Rational Homotopy Type, Memoirs A.M.S. 179 (1976).Google Scholar
  3. 3.
    Doeraene, J.-P.: L.S.-category in a model category, J. Pure Appl. Algebra 84 (1993), 215–261.Google Scholar
  4. 4.
    Dwyer, W. and Spalinski, J.: Homotopy theories and model categories, in I. M. James (ed.), Handbook of Algebraic Topology, North-Holland, 1995, pp. 73-126.Google Scholar
  5. 5.
    Elmendorf, A., Kriz, I., Mandell, M. and May, J.P.: Rings, Modules, and Algebras in Stable Homotopy Theory, Mathematical Surveys and Monographs, Vol. 47, American Mathematical Society, 1997.Google Scholar
  6. 6.
    Félix, Y. and Halperin, S.: Rational L.-S. category and its applications, Trans. A.M.S. 273 (1982), 1–37.Google Scholar
  7. 7.
    Félix, Y., Halperin, S. and Lemaire, J.-M.: Rational category and conelength of Poincaré complexes, Topology 37 (1998), 743–748.Google Scholar
  8. 8.
    Félix, Y., Halperin, S. and Thomas, J.-C.: Rational Homotopy Topology, Springer, Berlin, 2001.Google Scholar
  9. 9.
    Halperin, S. and Lemaire, J.-M.: Notions of category in differential algebra, in Algebraic Topology: Rational Homotopy, Springer Lecture Notes in Mathematics, Vol. 1318, pp. 138–153.Google Scholar
  10. 10.
    Hess, K.: A proof of Ganea's conjecture for rational spaces, Topology 30 (1991), 205–214.Google Scholar
  11. 11.
    Hovey, M.: Model Categories, Mathematical Surveys and Monographs, Vol. 63, American Mathematical Society, 1999.Google Scholar
  12. 12.
    Hovey, M.: Monoidal model categories, Trans. Amer. Math. Soc. (to appear), preprint available on the Hopf server Scholar
  13. 13.
    Hovey, M., Shipley, B. and Smith, J.: Symmetric spectra, J. Amer. Math. Soc. 13 (2000), 149–208.Google Scholar
  14. 14.
    Iwase, N.: Ganea's conjecture on Lusternik-Schnirelmann category, Bull. London Math. Soc. 30 (1998), 623–634.Google Scholar
  15. 15.
    Jessup, B.: Rational L.-S. category and a conjecture of Ganea, Trans. A.M.S. 317 (1990), 655–660.Google Scholar
  16. 16.
    Lydakis, M.: Smash products and _-spaces, Math. Proc. Cambridge Philos. Soc. 126 (1999), 311–328.Google Scholar
  17. 17.
    Quillen, D.: Homotopical Algebra, Springer Lecture Notes in Mathematics, Vol. 43, 1967.Google Scholar
  18. 18.
    Quillen, D.: Rational homotopy theory, Ann. Math. 90 (1969), 205–295.Google Scholar
  19. 19.
    Schwede, S. and Shipley, B.: Algebras and modules in monoidal model categories, Proc. London Math. Soc. (3) 80(2) (2000), 491–511.Google Scholar
  20. 20.
    Strøm, A.: The homotopy category is a homotopy category, Arch. Math. 23 (1972), 435–441.Google Scholar
  21. 21.
    Sullivan, D.: Infinitesimal Computations in Topology, Publ. IHES, Vol. 47, 1977, pp. 269–331.Google Scholar

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© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Kathryn Hess
    • 1
  1. 1.Département de mathématiquesEcole Polytechnique Fédérale de LausanneLausanneSwitzerland

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