Geodesic
The Verdier Hypercovering Theorem
Canadian mathematical bulletin, Tome 55 (2012) no. 2, pp. 319-328

Voir la notice de l'article provenant de la source Cambridge University Press

This note gives a simple cocycle-theoretic proof of the Verdier hypercovering theorem. This theorem approximates morphisms $[X,\,Y]$ in the homotopy category of simplicial sheaves or presheaves by simplicial homotopy classes of maps, in the case where $Y$ is locally fibrant. The statement proved in this paper is a generalization of the standard Verdier hypercovering result in that it is pointed (in a very broad sense) and there is no requirement for the source object $X$ to be locally fibrant.
DOI : 10.4153/CMB-2011-093-x
Mots-clés : 14F35, 18G30, 55U35, simplicial presheaf, hypercover, cocycle 
Jardine, J. F. The Verdier Hypercovering Theorem. Canadian mathematical bulletin, Tome 55 (2012) no. 2, pp. 319-328. doi: 10.4153/CMB-2011-093-x
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[1] [1] Artin, M. and Mazur, B, B., Etale Homotopy. Lecture Notes in Mathematics 100, Springer-Verlag, Berlin, 1969. Google Scholar

[2] [2] Blanc, D., Dwyer, W. G., and Goerss, P. G., The realization space of П-algebra: a moduli problem in algebraic topology. Topology 43(2004), no. 4, 857–892. Google Scholar | DOI

[3] [3] Brown, K. S., Abstract homotopy theory and generalized sheaf cohomology. Trans. Amer. Math. Soc. 186(1974), 419–458. Google Scholar | DOI

[4] [4] Dugger, D., Hollander, S., and Isaksen, D. C., Hypercovers and simplicial presheaves. Math. Proc. Cambridge Philos. Soc. 136(2004), no. 1, 9–51. doi=10.1017/S0305004103007175. Google Scholar | DOI

[5] [5] Dwyer, W. G. and Kan, D. M., Function complexes in homotopical algebra. Topology 19(1980), no. 4, 427–440. Google Scholar | DOI

[6] [6] Friedlander, E. M., Étale Homotopy of Simplicial Schemes. Annals of Mathematics Studies 104. Princeton University Press, Princeton, NJ, 1982. Google Scholar

[7] [7] Jardine, J. F., Simplicial objects in a Grothendieck topos. In: Applications of Algebraic K-Theory to Algebraic Geometry and Number Theory, Part I. Amererican Mathematical Society, Providence, RI, 1986, 193–239. Google Scholar

[8] [8] Jardine, J. F., Simplicial presheaves. J. Pure Appl. Algebra 47(1987), no. 1, 35–87. Google Scholar | DOI

[9] [9] Jardine, J. F., Universal Hasse-Witt classes. In: Algebraic K-Theory and Algebraic Number Theory. Contemp. Math. 83. American Mathematical Society, Providence, RI, 1989, pp. 83–100. Google Scholar

[10] [10] Jardine, J. F., Cocycle categories. In: Algebraic Topology 4. Springer, Berlin 2009, pp. 185–218. Google Scholar

[11] [11] Fabien, M., and Voevodsky, V., 1 -homotopy theory of schemes. Inst. Hautes études Sci. Publ. Math. 1999(2001), no. 90, 45–143. Google Scholar

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