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Locally constant functors

Published online by Cambridge University Press:  10 June 2009

DENIS–CHARLES CISINSKI
DENIS–CHARLES CISINSKI*
Affiliation:
LAGA, CNRS (UMR 7539), Université Paris 13, Avenue Jean-Baptiste Clément, 93430 VilletaneuseFrance. e-mail: [email protected]
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Abstract

We study locally constant coefficients. We first study the theory of homotopy Kan extensions with locally constant coefficients in model categories, and explain how it characterizes the homotopy theory of small categories. We explain how to interpret this in terms of left Bousfield localization of categories of diagrams with values in a combinatorial model category. Finally, we explain how the theory of homotopy Kan extensions in derivators can be used to understand locally constant functors.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 147 , Issue 3 , November 2009 , pp. 593 - 614
Copyright
Copyright © Cambridge Philosophical Society 2009

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References

REFERENCES

[Bar07]Barwick, C. On (enriched) left Bousfield localization of model categories, arXiv:0708.2067, (2007).Google Scholar
[Bek00]Beke, T.Sheafifiable homotopy model categories. Math. Proc. Camb. Phil. Soc. 129 (2000), 447–475.Google Scholar
[Bro73]Brown, K. S.Abstract homotopy and generalized sheaf cohomology. Trans. Amer. Math. Soc. 186 (1973), 419–458.CrossRefGoogle Scholar
[Cis03a]Cisinski, D.-C. Faisceaux localement asphériques, preprint. (2003). Available at the url address http://www.math.univ-paris13.fr/~cisinski/Google Scholar
[Cis03b]Cisinski, D.-C.Images directes cohomologiques dans les catégories de modèles. Ann. Math. Blaise Pascal 10 (2003), 195–244.CrossRefGoogle Scholar
[Cis06]Cisinski, D.-C. Les préfaisceaux comme modèles des types d'homotopie. Astérisque vol. 308, (2006).Google Scholar
[Cis08b]Cisinski, D.-C.Propriétés universelles et extensions de Kan dérivées. Theory and Appl. Categ. 20 (2008), no. 17, 605–649.Google Scholar
[DHI04]Dugger, D., Hollander, S. and Isaksen, D.Hypercovers and simplicial presheaves. Math. Proc. Camb. Phil. Soc. 136 (2004), no. 1, 9–51.Google Scholar
[DK80]Dwyer, W. G. and Kan, D. M.Calculating simplicial localizations. J. Pure Appl. Algebra 18 (1980), no. 1, 17–35.Google Scholar
[Dug01]Dugger, D.Combinatorial model categories have presentations. Adv. Math. 164 (2001), no. 1, 177–201.Google Scholar
[GZ67]Gabriel, P. and Zisman, M. Calculus of fractions and homotopy theory. Ergeb. Math. Grenzgeb. vol. 35 (Springer-Verlag, 1967).Google Scholar
[Hir03]Hirschhorn, P S. Model categories and their localizations. Math. Surveys Monog. vol. 99 (Amer. Math. Soc., 2003).Google Scholar
[Hov99]Hovey, M. Model categories. Math. Surveys Monog. vol. 63 (Amer. Math. Soc., 1999).Google Scholar
[JT07]Joyal, A. and Tierney, M. Quasi-categories vs Segal spaces. Categories in Algebra, Geometry and Physics. Contemp. Math. vol. 431 (Amer. Math. Soc., 2007), p. 277–326. Preprint version available as arXiv:math/0607820.Google Scholar
[Mal01]Maltsiniotis, G. Introduction à la théorie des dérivateurs (d'après Grothendieck). preprint, (2001).Google Scholar
[Mal05a]Maltsiniotis, G. La théorie de l'homotopie de Grothendieck. Astérisque vol. 301 (2005).Google Scholar
[Mal05b]Maltsiniotis, G.Structures d'asphéricité. Ann. Math. Blaise Pascal 12 (2005), no. 1, 1–39.Google Scholar
[Qui67]Quillen, D. Homotopical algebra. Lecture Notes in Mathematics, vol. 43 (Springer-Verlag, 1967).Google Scholar
[Qui73]Quillen, D. Higher algebraic K-theory. in Higher K-theories I. Lecture Notes in Mathematics, vol. 341 (Springer-Verlag, 1973), p. 85–147.Google Scholar
[RB06]Rădulescu-Banu, A. Cofibrations in homotopy theory. arXiv:math/0610009, (2006).Google Scholar
[Shu08]Shulman, M.Parametrized spaces model locally constant homotopy sheaves. Topology Appl. 155 (2008), no. 5, 412–432.Google Scholar
[Tab08]Tabuada, G.Higher K-theory via universal invariants. Duke Math. J. 145 (2008), no. 1, 121–206.Google Scholar
[Toë02]Toën, B.Vers une interprétation galoisienne de la théorie de l'homotopie. Cahiers Topologie Geom. Différentielle Categ. XLIII-3 (2002), 257–312.Google Scholar