Fibre sequences and localization of simplicial sheaves

Wendt, Matthias¹

¹Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstraße 1, D-79104, Freiburg im Breisgau, Germany

Algebraic and Geometric Topology, Tome 13 (2013) no. 3, pp. 1779-1813

Résumé

In this paper, we discuss the theory of quasifibrations in proper Bousfield localizations of model categories of simplicial sheaves. We provide a construction of fibrewise localization and use this construction to generalize a criterion for locality of fibre sequences due to Berrick and Dror Farjoun. The results allow a better understanding of unstable A1–homotopy theory.

Détail

DOI : 10.2140/agt.2013.13.1779

Classification : 55R65, 55P60, 18F20, 14F42
Keywords: Bousfield localization, simplicial sheaves, A^1-homotopy theory

Bibliographie

[1] A J Berrick, E D Farjoun, Fibrations and nullifications, Israel J. Math. 135 (2003) 205

[2] D Chataur, J Scherer, Fiberwise localization and the cube theorem, Comment. Math. Helv. 81 (2006) 171

[3] A Dold, R Thom, Quasifaserungen und unendliche symmetrische Produkte, Ann. of Math. 67 (1958) 239

[4] D Dugger, Combinatorial model categories have presentations, Adv. Math. 164 (2001) 177

[5] E D Farjoun, Cellular spaces, null spaces and homotopy localization, Lecture Notes in Mathematics 1622, Springer (1996)

[6] P G Goerss, J F Jardine, Localization theories for simplicial presheaves, Canad. J. Math. 50 (1998) 1048

[7] P G Goerss, J F Jardine, Simplicial homotopy theory, Progress in Mathematics 174, Birkhäuser (1999)

[8] P S Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, American Mathematical Society (2003)

[9] J Hornbostel, Localizations in motivic homotopy theory, Math. Proc. Cambridge Philos. Soc. 140 (2006) 95

[10] M Hovey, Model categories, Mathematical Surveys and Monographs 63, American Mathematical Society (1999)

[11] J F Jardine, Boolean localization, in practice, Doc. Math. 1 (1996) 245

[12] J F Jardine, Motivic symmetric spectra, Doc. Math. 5 (2000) 445

[13] F Morel, $\mathbb A^1$–algebraic topology over a field, Lecture Notes in Mathematics 2052, Springer (2012)

[14] F Morel, V Voevodsky, $\mathbf{A}^1$–homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. (1999) 45

[15] C Rezk, Fibrations and homotopy colimits of simplicial sheaves

[16] Y B Rudyak, On Thom spectra, orientability, and cobordism, Springer Monographs in Mathematics, Springer (1998)

[17] M Wendt, On fibre sequences in motivic homotopy theory, PhD thesis, Universität Leipzig (2007)

[18] M Wendt, Classifying spaces and fibrations of simplicial sheaves, J. Homotopy Relat. Struct. 6 (2011) 1

[19] M Wendt, Rationally trivial torsors in $\mathbb A^1$–homotopy theory, J. K-Theory 7 (2011) 541