Effective homology for homotopy colimit and cofibrant replacement

Marek Filakovský

Archivum Mathematicum (2014)

  • Volume: 050, Issue: 5, page 273-286
  • ISSN: 0044-8753

Abstract

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We extend the notion of simplicial set with effective homology presented in [22] to diagrams of simplicial sets. Further, for a given finite diagram of simplicial sets X : ℐ → sSet such that each simplicial set X ( i ) has effective homology, we present an algorithm computing the homotopy colimit hocolim X as a simplicial set with effective homology. We also give an algorithm computing the cofibrant replacement X cof of X as a diagram with effective homology. This is applied to computing of equivariant cohomology operations.

How to cite

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Filakovský, Marek. "Effective homology for homotopy colimit and cofibrant replacement." Archivum Mathematicum 050.5 (2014): 273-286. <http://eudml.org/doc/262212>.

@article{Filakovský2014,
abstract = {We extend the notion of simplicial set with effective homology presented in [22] to diagrams of simplicial sets. Further, for a given finite diagram of simplicial sets $X \colon \mathcal \{I\}\rightarrow \mbox\{sSet\}$ such that each simplicial set $X(i)$ has effective homology, we present an algorithm computing the homotopy colimit $\mbox\{hocolim\}\,X$ as a simplicial set with effective homology. We also give an algorithm computing the cofibrant replacement $X^\{\mbox\{cof\}\}$ of $X$ as a diagram with effective homology. This is applied to computing of equivariant cohomology operations.},
author = {Filakovský, Marek},
journal = {Archivum Mathematicum},
keywords = {homotopy colimit; cofibrant replacement; effective homology; equivariant; homotopy colimit; cofibrant replacement; effective homology; equivariant},
language = {eng},
number = {5},
pages = {273-286},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Effective homology for homotopy colimit and cofibrant replacement},
url = {http://eudml.org/doc/262212},
volume = {050},
year = {2014},
}

TY - JOUR
AU - Filakovský, Marek
TI - Effective homology for homotopy colimit and cofibrant replacement
JO - Archivum Mathematicum
PY - 2014
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 050
IS - 5
SP - 273
EP - 286
AB - We extend the notion of simplicial set with effective homology presented in [22] to diagrams of simplicial sets. Further, for a given finite diagram of simplicial sets $X \colon \mathcal {I}\rightarrow \mbox{sSet}$ such that each simplicial set $X(i)$ has effective homology, we present an algorithm computing the homotopy colimit $\mbox{hocolim}\,X$ as a simplicial set with effective homology. We also give an algorithm computing the cofibrant replacement $X^{\mbox{cof}}$ of $X$ as a diagram with effective homology. This is applied to computing of equivariant cohomology operations.
LA - eng
KW - homotopy colimit; cofibrant replacement; effective homology; equivariant; homotopy colimit; cofibrant replacement; effective homology; equivariant
UR - http://eudml.org/doc/262212
ER -

References

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  1. Bousfield, A.K., Kan, D.M., 10.1007/978-3-540-38117-4_12, Lecture Notes in Math., no. 304, Springer, 1972. (1972) Zbl0259.55004MR0365573DOI10.1007/978-3-540-38117-4_12
  2. Bredon, G., Equivariant cohomology theories, Lecture Notes in Math., no. 34, Springer-Verlag, Berlin-New York, 1967. (1967) Zbl0162.27202MR0214062
  3. Brown, R., The twisted Eilenberg-Zilber theorem, 1965 Simposio di Topologia (Messina, 1964), 1965, pp. 33–37. (1965) MR0220273
  4. Čadek, M., Krčál, M., Matoušek, J., Sergeraert, F., Vokřínek, L., Wagner, U., 10.1145/2597629, J. ACM 61 (2014), article no. 17. (2014) Zbl1295.68196MR3215297DOI10.1145/2597629
  5. Čadek, M., Krčál, M., Matoušek, J., Vokřínek, L., Wagner, U., 10.1137/120899029, SIAM J. Comput. 43 (2014), 1728–1780. (2014) MR3268623DOI10.1137/120899029
  6. Čadek, M., Krčál, M., Vokřínek, L., Algorithmic solvability of lifting extension problem, arXiv:1307.6444, 2013. 
  7. Dugger, D., A primer on homotopy colimits, http://math.uoregon.edu/~ddugger/hocolim.pdf, 2008. 
  8. Dwyer, W., Hirschhorn, P., Kan, D., Smith, J., Homotopy Limit Functors on Model Categories and Homotopical Categories, Math. Surveys Monogr., vol. 113, American Mathematical Society, Providence, 2004. (2004) Zbl1072.18012MR2102294
  9. Dwyer, W.G., Kan, D.M., 10.1016/1385-7258(84)90015-5, Nederl. Akad. Wetensch. Indag. Math. 46 (2) (1984), 139–146. (1984) Zbl0555.55018MR0749527DOI10.1016/1385-7258(84)90015-5
  10. Eilenberg, S., MacLane, S., On the groups H ( π , n ) , I, , Ann. of Math. (2) 58 (1953), 55–106. (1953) Zbl0050.39304MR0056295
  11. Eilenberg, S., MacLane, S., On the groups H ( π , n ) , II, , Ann. of Math. (2) 60 (1954), 49–139. (1954) Zbl0055.41704MR0065163
  12. Elmendorf, A.D., 10.1090/S0002-9947-1983-0690052-0, Trans. Amer. Math. Soc. 277 (1983), 275–284. (1983) MR0690052DOI10.1090/S0002-9947-1983-0690052-0
  13. Filakovský, M., Vokřínek, L., Are two maps homotopic? An algorithmic viewpoint, arXiv:1312.2337, 2013. 
  14. Goerss, P.G., Jardine, J.F., Simplicial homotopy theory, Birkhauser, Boston-Basel-Berlin, 1999. (1999) Zbl0949.55001MR1711612
  15. Gugenheim, V.K.A.M., On the chain-complex of a fibration, Illinois J. Math. 16 (1972), 398–414. (1972) Zbl0238.55015MR0301736
  16. Heras, J., Effective homology for the pushout of simplicial sets, Proceedings XII Encuentros de Algebra Computacional y Aplicaciones (EACA 2010), 2010, pp. 152–156. (2010) 
  17. Isaacson, S.B., Exercises on homotopy colimits, http://math.mit.edu/~mbehrens/TAGS/Isaacson_exer.pdf. 
  18. Krčál, M., Matoušek, J., Sergeraert, F., Polynomial-time homology for simplicial Eilenberg-MacLane spaces, arXiv:1201.6222, 2012. Zbl1295.68201MR3124946
  19. May, J.P., Simplicial Objects in Algebraic Topology, Chicago Lectures in Math., Univ. Chicago Press, 1992, 1992 reprint of 1967 original. (1992) Zbl0769.55001MR1206474
  20. May, J.P., Piacenza, R.J., Cole, M., Equivariant homotopy and cohomology theory: Dedicated to the memory of Robert J. Piacenza, Providence, R.I.: Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, 1996. (1996) MR1413302
  21. Riehl, E., Categorical Homotopy Theory, Cambridge University Press, 2014. (2014) MR3221774
  22. Rubio, J., Sergeraert, F., Constructive Homological Algebra and Applications, Tech. report, Written in 2006 for a MAP Summer School at the University of Genova, 2012, arXiv:1208.3816v2. (2012) 
  23. Shih, W., Homologie des espaces fibrés, Publications Mathématiques de l'IHÉS 13 (1962), 5–87. (1962) Zbl0105.16903MR0144348
  24. Stephan, M., On equivariant homotopy theory for model categories, arXiv:1308.0856, 2013. 

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