Journal of Homotopy and Related Structures

, Volume 11, Issue 4, pp 679–713 | Cite as

2-track algebras and the Adams spectral sequence

  • Hans-Joachim Baues
  • Martin FranklandEmail author


In previous work of the first author and Jibladze, the \(E_3\)-term of the Adams spectral sequence was described as a secondary derived functor, defined via secondary chain complexes in a groupoid-enriched category. This led to computations of the \(E_3\)-term using the algebra of secondary cohomology operations. In work with Blanc, an analogous description was provided for all higher terms \(E_r\). In this paper, we introduce 2-track algebras and tertiary chain complexes, and we show that the \(E_4\)-term of the Adams spectral sequence is a tertiary Ext group in this sense. This extends the work with Jibladze, while specializing the work with Blanc in a way that should be more amenable to computations.


Adams spectral sequence Tertiary Toda bracket 2-track groupoid Bigroupoid Double groupoid 

Mathematics Subject Classification

55T15 18G50 55S20 



We thank the referee for their helpful comments. The second author thanks the Max-Planck-Institut für Mathematik Bonn for its generous hospitality, as well as David Blanc, Robert Bruner, Dan Christensen, and Dan Isaksen for useful conversations.


  1. 1.
    Adams, J.F.: On the structure and applications of the Steenrod algebra. Comment. Math. Helv. 32, 180–214 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baues, H.-J.: The Algebra of Secondary Cohomology Operations, Progress in Mathematics, vol. 247. Birkhäuser Verlag, Basel (2006)zbMATHGoogle Scholar
  3. 3.
    Baues, H.-J., Jibladze, M.: Secondary derived functors and the Adams spectral sequence. Topology 45(2), 295–324 (2006). doi: 10.1016/ MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Baues, H.J., Blanc, D.: Stems and spectral sequences. Algebr. Geom. Topol. 10(4), 2061–2078 (2010). doi: 10.2140/agt.2010.10.2061
  5. 5.
    Baues, H.-J., Jibladze, M.: Dualization of the Hopf algebra of secondary cohomology operations and the Adams spectral sequence. J. K-Theory 7(2), 203–347 (2011). doi: 10.1017/is010010029jkt133 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Baues, H.-J., Blanc, D.: Higher order derived functors and the Adams spectral sequence. J. Pure Appl. Algebra 219(2), 199–239 (2015). doi: 10.1016/j.jpaa.2014.04.018 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Blanc, D., Paoli, S.: Two-track categories. J. K-Theory 8(1), 59–106 (2011). doi: 10.1017/is010003020jkt116 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bousfield, A.K., Kan, D.M.: The homotopy spectral sequence of a space with coefficients in a ring. Topology 11, 79–106 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bousfield, A.K., Friedlander, E.M.: Homotopy theory of a spaces, spectra, and bisimplicial sets, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977): II, Lecture Notes in Math., vol. 658, pp. 80–130. Springer, Berlin (1978)Google Scholar
  10. 10.
    Brown, R., Hardie, K.A., Kamps, K.H., Porter, T.: A homotopy double groupoid of a Hausdorff space. Theory Appl. Categ. 10, 71–93 (2002)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Cegarra, A.M., Heredia, B.A., Remedios, J.: Double groupoids and homotopy 2-types. Appl. Categ. Struct. 20(4), 323–378 (2012). doi: 10.1007/s10485-010-9240-1 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hardie, K.A., Kamps, K.H., Kieboom, R.W.: A homotopy bigroupoid of a topological space. Appl. Categ. Struct. 9(3), 311–327 (2001). doi: 10.1023/A:1011270417127 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mandell, M.A., May, J.P., Schwede, S., Shipley, B.: Model categories of diagram spectra. Proc. Lond. Math. Soc. (3) 82(2), 441–512 (2001). doi: 10.1112/S0024611501012692
  14. 14.
    Maunder, C.R.F.: On the differentials in the Adams spectral sequence. Proc. Camb. Philos. Soc. 60, 409–420 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Tamsamani, Z.: Sur des notions de n-catégorie et n-groupoïde non strictes via des ensembles multi-simpliciaux. K-Theory 16(1), 51–99 (1999). doi: 10.1023/A:1007747915317 (French, with English summary)

Copyright information

© Tbilisi Centre for Mathematical Sciences 2016

Authors and Affiliations

  1. 1.Max-Planck-Institut für MathematikBonnGermany
  2. 2.Institut für MathematikUniversität OsnabrückOsnabrückGermany

Personalised recommendations