Coherent homotopy and localization

  • Sibe Mardešić
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

It is well known that localization of the category Top at homotopy equivalences yields the homotopy category H(Top) (see Theorem 4.35). The analogous procedure for inverse systems, indexed by Λ, is the localization of the category Top Λ at level homotopy equivalences. The resulting category will be denoted by Ho(Top Λ ). One also considers the corresponding localization of pro -Top and denotes it by Ho(pro-Top). The main results of this section show that the localized categories, obtained in this manner, are isomorphic to the corresponding coherent homotopy categories CH(Top Λ ) and CH(pro-Top), respectively. Therefore, the coherent homotopy categories can be viewed as concrete realizations of the rather abstract localized categories. The first subsection is devoted to an isomorphism theorem in coherent homotopy, which enables one to obtain functors between the categories in question. The second subsection defines cotelescopes (homotopy limits), a tool needed to prove that these functors are isomorphisms of categories.

Keywords

Coherence Verse 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliographic notes

  1. Mardesié, S. (1999a): Coherent homotopy and localization. Topology Appl. 94, 253–274MathSciNetCrossRefGoogle Scholar
  2. Mardesié, S., Segal, J. (1982): Shape theory. The inverse system approach. North - Holland, AmsterdamGoogle Scholar
  3. Milnor, J. (1962): On axiomatic homology theory. Pacific J. Math. 12, 337–341MathSciNetMATHGoogle Scholar
  4. Edwards, D.A., Hastings, H.M. (1976a): Cech and Steenrod homotopy theories with applications to Geometric Topology. Lecture Notes in Math. 542, Springer, Berlin Heidelberg New YorkGoogle Scholar
  5. Segal, G.B. (1968): Classifying spaces and spectral sequences. Publ. Math. IHES 34, 105–112MATHGoogle Scholar
  6. Bousfield, A.K., Kan, D.M. (1972): Homotopy limits, completions and localizations. Lecture Notes in Math. 304, Springer, Berlin Heidelberg New YorkCrossRefGoogle Scholar
  7. Boardman, J.M., Vogt, R.M. (1973): Homotopy invariant algebraic structures on topological spaces. Lecture Notes in Math. 347, Springer, Berlin Heidelberg New YorkGoogle Scholar
  8. Vogt, R.M. (1973): Homotopy limits and colimits. Math. Z. 134, 11–52MathSciNetMATHCrossRefGoogle Scholar
  9. Segal, G.B. (1974): Categories and cohomology theories. Topology 13, 293–312MathSciNetMATHCrossRefGoogle Scholar
  10. Lisitsa, Yu.T. (1982b): Cotelescopes and the Kuratowski-Dugundji theorem in shape theory. Dokl. Akad. Nauk SSSR 265, No. 5, 1064–1068 (Russian) (Soviet Math. Dokl. 26, No. 1, 205–210 )MATHGoogle Scholar
  11. Lisica, Ju.T., Mardesié, S. (1984b): Coherent prohomotopy and strong shape theory. Glasnik Mat. 19, 335–399Google Scholar
  12. Thiemann, H. (1995): Strong shape and fibrations. Glasnik Mat. 30, 135–174MathSciNetMATHGoogle Scholar
  13. Boardman, J.M., Vogt, R.M. (1973): Homotopy invariant algebraic structures on topological spaces. Lecture Notes in Math. 347, Springer, Berlin Heidelberg New YorkGoogle Scholar
  14. Edwards, D.A., Hastings, H.M. (1976a): Cech and Steenrod homotopy theories with applications to Geometric Topology. Lecture Notes in Math. 542, Springer, Berlin Heidelberg New YorkGoogle Scholar
  15. Quillen, D.G. (1967): Homotopical algebra. Lecture Notes in Math. 43, Springer, Berlin Heidelberg New YorkGoogle Scholar
  16. Porter, T. (1974): Stability results for topological spaces. Math. Z. 140, 1–21MathSciNetMATHCrossRefGoogle Scholar
  17. Porter, T. (1988): On the two definitions of Ho(pro-C). Topology Appl. 28, 283–293CrossRefGoogle Scholar
  18. Cordier, J.M., Porter, T. (1986): Vogt’s theorem on categories of homotopy coherent diagrams. Math. Proc. Cambridge Phil. Soc. 100, 65–90MathSciNetMATHCrossRefGoogle Scholar
  19. Cordier, J.M. (1989): Comparaison de deux cathégories d’homotopie de morphismes cohérents. Cahiers Topol. Géom. Diff. 30, 257–275MathSciNetMATHGoogle Scholar
  20. Günther, B. (1991a): Comparison of the coherent pro-homotopy theories of Edwards–Hastings, Lisica- Mardes“ie and Günther. Glasnik Mat. 26, 141–176MATHGoogle Scholar
  21. Batanin, M.A. (1993): Coherent categories with respect to monads and coherent prohomotopy theory. Cahiers Topol. Géom. Diff. Categor. 34, No. 4, 279–304MathSciNetMATHGoogle Scholar
  22. Sekutkovski, N. (1997): Equivalence of coherent theories. Topology Appl. 75, 113–123.MathSciNetCrossRefGoogle Scholar
  23. Cordier, J.M., Porter, T. (1986): Vogt’s theorem on categories of homotopy coherent diagrams. Math. Proc. Cambridge Phil. Soc. 100, 65–90MathSciNetMATHCrossRefGoogle Scholar
  24. Hardie, K.A, Kamps, K.H. (1989): Track homotopy over a fixed space. Glasnik Mat. 24, 161–179MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Sibe Mardešić
    • 1
  1. 1.Department of MathematicsUniversity of ZagrebZagrebCroatia

Personalised recommendations