Applied Categorical Structures

, Volume 2, Issue 4, pp 351–406 | Cite as

Homotopical algebra in homotopical categories

  • Marco Grandis


We develop here a version of abstract homotopical algebra based onhomotopy kernels andcokernels, which are particular homotopy limits and colimits. These notions are introduced in anh-category, a sort of two-dimensional context more general than a 2-category, abstracting thenearly 2-categorical properties of topological spaces, continuous maps and homotopies. A setting which applies also, at different extents, to cubical or simplicial sets, chain complexes, chain algebras, ... and in which homotopical algebra can be established as a two-dimensional enrichment of homological algebra.

Actually, a hierarchy of notions ofh-,h1-, ...h4-categories is introduced, through progressive enrichment of thevertical structure of homotopies, so that the strongest notion,h4-category, is a sort of relaxed 2-category. After investigating homotopy pullbacks and homotopical diagrammatical lemmas in these settings, we introduceright semihomotopical categories, ash-categories provided with terminal object and homotopy cokernels (mapping cones), andright homotopical categories, provided also with anh4-structure and verifying second-order regularity properties forh-cokernels.

In these frames we study the Puppe sequence of a map, its comparison with the sequence of iterated homotopy cokernels and theh-cogroup structure of the suspension endofunctor. Left (semi-) homotopical categories, based on homotopy kernels, give the fibration sequence of a map and theh-group of loops. Finally, the self-dual notion of homotopical categories is considered, together with their stability properties.

Mathematics Subject Classifications (1991)

55U35 18D05 18G55 55P 55R05 

Key words

Homotopical algebra abstract homotopy theory 2-categories homotopy homotopy (co-)limits homotopy (co-)kernels mapping cone suspension Puppe sequence (co-)fibrations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    D. W. Anderson: Axiomatic homotopy theory, in:Algebraic Topology, Waterloo, 1978, Lect. Notes in Math.741, Springer-Verlag, 1979, pp. 520–547.Google Scholar
  2. 2.
    H. J. Baues:Algebraic Homotopy, Cambridge Univ. Press, 1989.Google Scholar
  3. 3.
    J. Bénabou: Some remarks on 2-categorical algebra (Part I),Bull. Soc. Math. Belgique 41 (1989), 127–194.Google Scholar
  4. 4.
    K. S. Brown: Abstract homotopy theory and generalised sheaf cohomology,Trans. Amer. Math. Soc. 186 (1973), 419–458.Google Scholar
  5. 5.
    A. Dold and D. Puppe: Homologie nicht-additiver Funktoren, Anwerdungen,Ann. Inst. Fourier Grenoble 11 (1961), 201–312.Google Scholar
  6. 6.
    M. Grandis: On the categorical foundations of homological and homotopical algebra,Cahiers Top. Géom. Diff. Catég. 33 (1992), 135–175.Google Scholar
  7. 7.
    M. Grandis:Homological Algebra in Non Abelian Settings, in preparation.Google Scholar
  8. 8.
    P. Gabriel and M. Zisman:Calculus of Fractions and Homotopy Theory, Springer-Verlag, 1967.Google Scholar
  9. 9.
    A. Heller: Stable homotopy categories,Bull. Amer. Math. Soc. 74 (1968), 28–63.Google Scholar
  10. 10.
    A. Heller:Homotopy Theories, Mem. Amer. Math. Soc.383 (1988).Google Scholar
  11. 11.
    R. Hartshorne:Residues and Duality, Lect. Notes in Math.20, Springer-Verlag, 1966.Google Scholar
  12. 12.
    P. J. Huber: Homotopy theories in general categories,Math. Ann. 144 (1961), 361–385.Google Scholar
  13. 13.
    P. J. Huber: Standard constructions in abelian categories,Math. Ann. 146 (1962), 321–325.Google Scholar
  14. 14.
    D. M. Kan: Abstract homotopy I–IV,Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 1092–1096;42 (1956), 255–258; 419–421; 542–544.Google Scholar
  15. 15.
    D. M. Kan: A combinatorial definition of homotopy groups,Ann. of Math. 61 (1958), 288–312.Google Scholar
  16. 16.
    K. H. Kamps: Faserungen und Cofaserungen in Kategorien mit Homotopiesystem, Dissertation, Saarbrücken 1968, 1–142.Google Scholar
  17. 17.
    K. H. Kamps: Über einige formale Eigenschaften von Faserungen undh-Faserungen,Manuscripta Math. 3 (1970), 237–255.Google Scholar
  18. 18.
    K. H. Kamps: Zur Homotopietheorie von Gruppoiden,Arch. Math. (Basel) 23 (1972), 610–618.Google Scholar
  19. 19.
    H. Kleisli: Homotopy theory in abelian categories,Canad. J. Math. 14 (1962), 139–169.Google Scholar
  20. 20.
    S. Mac Lane:Categories for the Working Mathematician, Springer-Verlag, 1971.Google Scholar
  21. 21.
    H. J. Marcum: Homotopy equivalences in 2-categories, in:Groups of Self-Equivalences and Related Topics, Montréal 1988, Lect. Notes in Math.1425, Springer-Verlag, 1990, pp. 71–86.Google Scholar
  22. 22.
    M. Mather: Pull-backs in homotopy theory,Can. J. Math. 28 (1976), 225–263.Google Scholar
  23. 23.
    T. Müller: Zur Theorie der Würfelsätze, Dissertation, Hagen, 1982.Google Scholar
  24. 24.
    D. Puppe: Homotopiemengen und ihre induzierten Abbildungen, I,Math. Z. 69 (1958), 299–344.Google Scholar
  25. 25.
    D. Puppe: On the formal structure of stable homotopy theory, in:Colloquium on Algebraic Topology, Math. Inst., Aarhus Univ., 1962, 65–71.Google Scholar
  26. 26.
    D. G. Quillen:Homotopical Algebra, Lect. Notes in Math.43, Springer-Verlag, 1967.Google Scholar
  27. 27.
    S. Rodriguez: Homotopía en categorías aditivas,Rev. Acad. Cienc. Zaragoza 43 (1988), 77–92.Google Scholar
  28. 28.
    R. Street: Categorical structures, in:Handbook of Algebra, Vol. 2, Elsevier Science Publishers, North Holland, to appear.Google Scholar
  29. 29.
    J. L. Verdier:Catégories Dérivées, Séminaire de Géometrie algébrique du Bois Marie SGA 4 1/2, Cohomologie étale, Lect. Notes in Math.569, Springer-Verlag, 1977, 262–311.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Marco Grandis
    • 1
  1. 1.Dipartimento di MatematicaUniversità di GenovaGenovaItaly

Personalised recommendations