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Model structures for pro-simplicial presheaves

Published online by Cambridge University Press:  24 May 2011

J.F. Jardine
J.F. Jardine
Affiliation:
Department of Mathematics, University of Western Ontario, London, Ontario N6A5B7, Canadajardine@uwo.ca
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Abstract

This paper displays model structures for the category of pro-objects in simplicial presheaves on an arbitrary small Grothendieck site. The first of these is an analogue of the Edwards-Hastings model structure for pro-simplicial sets, in which the cofibrations are monomorphisms and the weak equivalences are specified by comparisons of function complexes. Other model structures are built from the Edwards-Hastings structure by using Bousfield-Friedlander localization techniques. There is, in particular, an n-type structure for pro-simplicial presheaves, and also a model structure in which the map from a pro-object to its Postnikov tower is formally inverted.

Keywords

simplicial presheavespro-objectsn-typesPostnikov tower

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Type
Research Article
Information
Journal of K-Theory , Volume 7 , Issue 3 , June 2011 , pp. 499 - 525
Copyright
Copyright © ISOPP 2011

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References

1.Artin, M. and Mazur, B.. Etale homotopy. Lecture Notes in Math. 100, Springer-Verlag, Berlin, 1969.Google Scholar
2.Biedermann, Georg. On the homotopy theory of n-types. Homology, Homotopy Appl. 10(1):305–325, 2008.CrossRefGoogle Scholar
3.Bousfield, A. K.. On the telescopic homotopy theory of spaces. Trans. Amer. Math. Soc. 353(6):2391–2426 (electronic), 2001.Google Scholar
4.Bousfield, A. K. and Friedlander, E. M.. Homotopy theory of Γ-spaces, spectra, and bisimplicial sets. In Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II Lecture Notes in Math. 658, pages 80–130. Springer, Berlin, 1978.Google Scholar
5.Edwards, David A. and Hastings, Harold M.. Čech and Steenrod homotopy theories with applications to geometric topology. Lecture Notes in Math. 542, Springer-Verlag, Berlin, 1976.Google Scholar
6.Fausk, Halvard and Isaksen, Daniel C.. Model structures on pro-categories. Homology, Homotopy Appl. 9(1):367–398, 2007.CrossRefGoogle Scholar
7.Isaksen, Daniel C.. Strict model structures for pro-categories. In Categorical decomposition techniques in algebraic topology (Isle of Skye, 2001), Progr. Math. 215, pages 179–198. Birkhäuser, Basel, 2004.Google Scholar
8.Jardine, J. F.. Generalized Étale Cohomology Theories, Progress in Mathematics 146, Birkhäuser Verlag, Basel, 1997.Google Scholar
9.Jardine, J. F.. Fibred sites and stack cohomology. Math. Z. 254(4):811–836, 2006.CrossRefGoogle Scholar
10.Jardine, J. F.. Intermediate model structures for simplicial presheaves. Canad. Math. Bull. 49(3):407–413, 2006.CrossRefGoogle Scholar
11.Jardine, J.F.. Galois descent criteria. Preprint, http://www.math.uwo.ca/~jardine/papers/preprints, 2010.Google Scholar
12.Kashiwara, Masaki and Schapira, Pierre. Categories and sheaves, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 332, Springer-Verlag, Berlin, 2006.Google Scholar