Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-05-14T10:04:38.451Z Has data issue: false hasContentIssue false
  • English
  • Français

Stable Homotopy Theory of Simplicial Presheaves

Published online by Cambridge University Press:  20 November 2018

J. F. Jardine
J. F. Jardine*
Affiliation:
University of Western Ontario, London, Ontario
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let C be an arbitrary Grothendieck site. The purpose of this note is to show that, with the closed model structure on the category S Pre(C) of simplicial presheaves in hand, it is a relatively simple matter to show that the category S Pre(C)stab of presheaves of spectra (of simplicial sets) satisfies the axioms for a closed model category, giving rise to a stable homotopy theory for simplicial presheaves. The proof is modelled on the corresponding result for simplicial sets which is given in [1], and makes direct use of their Theorem A.7.

This result gives a precise description of the associated stable homotopy category Ho(S Pre(C))stab, according to well known results of Quillen [6]. One will recall, however, that it is preferable to have several different descriptions of the stable homotopy category, for the construction of smash products and the like.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 3 , 01 June 1987 , pp. 733 - 747
Copyright
Copyright © Canadian Mathematical Society 1987

References

1. Bousfield, A. K. and Friedlander, E. M., Homotopy theory of Γ-spaces, spectra, and bisimplicial sets, Springer Lecture Notes in Math. 658 (1978), 80–150.Google Scholar
2. Brown, K. S., Abstract homotopy theory and generalized sheaf cohomology, Trans. A.M.S. 186 (1973), 419–458.Google Scholar
3. Dwyer, W. and Friedlander, E., Algebraic and étale K-theory, Trans. AMS 292 (1985), 247–280.Google Scholar
4. Gabriel, P. and Zisman, M., Calculus of fractions and homotopy theory (Springer-Verlag, New York, 1967).CrossRefGoogle Scholar
5. Jardine, J. F., Simplicial presheaves to appear in J. Pure Applied Algebra.Google Scholar
6. Quillen, D., Homotopical algebra, Springer Lecture Notes in Math. 43 (1967).CrossRefGoogle Scholar
7. Quillen, D., The geometric realization of a Kan fibration is a Serre fibration, Proc. AMS 19 (1968), 1499–1500.Google Scholar
8. Thomason, R., Algebraic K-theory and étale cohomology, Ann. Scient. Éc. Norm. Sup., 4e série 75 (1985), 437–552.Google Scholar