Skip to main content
SpringerLink
Log in
Book cover

Algebraic Topology Homotopy and Group Cohomology pp 146–185Cite as

Categorical models of N-types for pro-crossed complexes and ℑn-prospaces

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1509))

Abstract

As part of a program to study Pro n-types and Proper n-types of locally finite simplicial complexes, in this paper we give notions of n-fibrations, n-cofibrations and weak n-equivalences in the category of crossed complexes Crs that satisfy the axioms for a closed model structure in the sense of Quillen. The category obtained by formal inverting the weak n-equivalences Hon(Crs) is said to be the category of n-types of crossed complexes. We also extend the notions above to pro-crossed complexes, pro-simplicial sets and prospaces to obtain the categories Hon(proCrs), Hon(proSS) and Hon(proTop).

We consider the notions of ℑn-space (a slight modification of the notion of Jn-space given by J.H.C. Whitehead), ℑn-crossed complex and their generalizations to the categories of prospaces and pro-crossed complexes and we prove that the category of n-types of ℑn-prospace is equivalent to the category of n-types of ℑn-pro-crossed complexes. WE also use the properties of skeleton, coskeleton and truncation functors to compare categories of n-types to categories of homotopy types. This analysis allows one to give new category models for the categories of n-types of prospaces and pro-crossed complexes.

The authors acknowledge the finantial help given by the British-Spanish joint research program British Council-M.E.C., 1988-89, 51/18' and the research project P587-0062 of the DGYCYT.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   59.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Artin and B. Mazur, "Étale Homotopy", Lect. Notes in Math., no 100, Springer, 1969.

    Google Scholar 

  2. D. Edwards and H. Hastings, "Čech and Steenrod homotopy theories with applications to Geometric Topology", Lect. Notes Math., 542, Springer, 1976.

    Google Scholar 

  3. H.J. Baues, "Combinatorial Homotopy and 4-dimensional Complexes", deGruyter Expositions in Mathematics Vol. 2, deGruyter Berlin-New York, 1991.

    Google Scholar 

  4. H.J. Baues, "Algebraic Homotopy", Cambridge Studies in advanced mathematics 15, Cambridge University Press, 1989.

    Google Scholar 

  5. R. Brown and M. Golasiński, "The closed model category of crossed complexes", Cah. Top. Geom. Diff. Cat. 30 (1989) 61–82.

    MATH  Google Scholar 

  6. R. Brown and P.J. Higgins, "Tensor products and homotopies for ω-groupoids and crossed complexes", J. Pure Appl. Algebra 47 (1987) 1–33.

    Article  MathSciNet  MATH  Google Scholar 

  7. R. Brown and P.J. Higgins, "The classifying space of a crossed module", U.C.N.W. Maths. Preprint 90.20 (36p).

    Google Scholar 

  8. R. Brown and P.J. Higgins, "Crossed complexes and chain complexes with operators", Math. Proc. Camb. Phil. Soc. 107 (1990) 33–57.

    Article  MathSciNet  MATH  Google Scholar 

  9. P. Carrasco and A.M. Cegarra, "Group theoretic models for homotopy types", J. Pure Appl. Algebra, (1989).

    Google Scholar 

  10. P. Gabriel and M. Zisman, "Calculus of fractions and Homotopy Theory", Springer, Berlin (1966).

    MATH  Google Scholar 

  11. A. Grothendieck, "Technique De Descente Et Théorèmes d'Existence En Géométrie Algèbrique I–IV", Seminar Bourbaki, Exp. 190, 195, 212, 221, 1959–60, 1960–61.

    Google Scholar 

  12. L.J. Hernández and T. Porter, "Proper pointed maps from ℝn+1 to a σ-compact space", Math. Proc. Camb. Phil. Soc. 103 (1988) 462–475.

    Article  MathSciNet  MATH  Google Scholar 

  13. L.J. Hernández and T. Porter, "Global analogues of the Brown-Crossman proper homotopy groups", Math. Proc. Camb. Phil. Soc. 104 (1988) 483–496.

    Article  MATH  Google Scholar 

  14. L.J. Hernández and T. Porter, "An embedding theorem for proper n-types at infinity", preprint (1990).

    Google Scholar 

  15. S. Mac Lane, "Historical note", J. Algebra 60 (1979) 319–320.

    Article  Google Scholar 

  16. S. Mac Lane and J.H.C. Whitehead, "On the 3-type of a complex", Proc., Nat., Acad., Sci., Washington, 36, (1950) 41–48.

    Article  MathSciNet  Google Scholar 

  17. T. Porter, "Čech homotopy I", J. London Math. Soc. 6 (1963) 429–436.

    MATH  Google Scholar 

  18. D. Quillen, "Homotopical Algebra", Lect. Notes in Math, 43, Springer, 1967.

    Google Scholar 

  19. A. Strøm,"The homotopy category is a homotopy category", Arch. Math. 23 (1973) 435–441.

    Article  MathSciNet  MATH  Google Scholar 

  20. J.H.C. Whitehead, "Combinatorial homotopy. I", Bull. Amer. Math. Soc., 55(1949) 213–245.

    Article  MathSciNet  MATH  Google Scholar 

  21. J.H.C. Whitehead, "Combinatorial homotopy, II", Bull. Amer. Math. Soc., 55(1949) 453–496.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matcmáticas, Universidad de Zaragoza, 50009, Zaragoza, Spain

    L. J. Hernández

  2. School of Mathematics, University College of North Wales, Dean Street, Bangor, LL57 1UT, Gwynedd, U.K.

    T. Porter

Authors
  1. L. J. Hernández
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. Porter
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Jaume Aguadé Manuel Castellet Frederick Ronald Cohen

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer-Verlag

About this paper

Cite this paper

Hernández, L.J., Porter, T. (1992). Categorical models of N-types for pro-crossed complexes and ℑn-prospaces. In: Aguadé, J., Castellet, M., Cohen, F.R. (eds) Algebraic Topology Homotopy and Group Cohomology. Lecture Notes in Mathematics, vol 1509. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0087509

Download citation

  • DOI: https://doi.org/10.1007/BFb0087509

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55195-9

  • Online ISBN: 978-3-540-46772-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation