An extension of Quillen’s Theorem B

Moerdijk, Ieke¹ ; Nuiten, Joost²

¹Mathematical Institute, Utrecht University, Utrecht, Netherlands
²Institut Montpelliérain Alexander Grothendieck, Université de Montpellier, Montpellier, France

Algebraic and Geometric Topology, Tome 20 (2020) no. 4, pp. 1769-1794

Résumé

We prove a general version of Quillen’s Theorem B, for actions of simplicial categories, in an arbitrary left Bousfield localization of the homotopy theory of simplicial presheaves over a site. As special cases, we recover a version of the group completion theorem in this general context, as well a version of Puppe’s theorem on the stability of homotopy colimits in an ∞–topos, due to Rezk.

Détail

DOI : 10.2140/agt.2020.20.1769

Classification : 18F20, 55U35, 18G30, 55U10
Keywords: Quillen's Theorem B, group completion, simplicial presheaf, Nisnevich site, Bousfield localization, Rezk descent

Bibliographie

[1] A K Bousfield, The localization of spaces with respect to homology, Topology 14 (1975) 133 | DOI

[2] K S Brown, Abstract homotopy theory and generalized sheaf cohomology, Trans. Amer. Math. Soc. 186 (1973) 419 | DOI

[3] W Chachólski, W Pitsch, J Scherer, Homotopy pull-back squares up to localization, from: "An alpine anthology of homotopy theory" (editors D Arlettaz, K Hess), Contemp. Math. 399, Amer. Math. Soc. (2006) 55 | DOI

[4] D C Cisinski, Locally constant functors, Math. Proc. Cambridge Philos. Soc. 147 (2009) 593 | DOI

[5] P Deligne, Critère d’existence de points,

[6] B Harris, Bott periodicity via simplicial spaces, J. Algebra 62 (1980) 450 | DOI

[7] J F Jardine, Simplicial presheaves, J. Pure Appl. Algebra 47 (1987) 35 | DOI

[8] J F Jardine, Diagrams and torsors, –Theory 37 (2006) 291 | DOI

[9] A Joyal, Lettre d’André Joyal à Alexandre Grothendieck, (1984)

[10] J Lurie, Higher topos theory, 170, Princeton Univ. Press (2009) | DOI

[11] M Makkai, G E Reyes, First order categorical logic: model-theoretical methods in the theory of topoi and related categories, 611, Springer (1977) | DOI

[12] D Mcduff, G Segal, Homology fibrations and the “group-completion” theorem, Invent. Math. 31 (1976) 279 | DOI

[13] I Moerdijk, Bisimplicial sets and the group-completion theorem, from: "Algebraic –theory : connections with geometry and topology" (editors J F Jardine, V P Snaith), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 279, Kluwer (1989) 225 | DOI

[14] F Morel, V Voevodsky, A1–homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999) 45 | DOI

[15] V Puppe, A remark on “homotopy fibrations”, Manuscripta Math. 12 (1974) 113 | DOI

[16] D Quillen, On the group completion of a simplicial monoid | DOI

[17] D Quillen, Higher algebraic K–theory, I, from: "Algebraic –theory, I : Higher –theories" (editor H Bass), Lecture Notes in Math. 341, Springer (1973) 85 | DOI

[18] C Rezk, Fibrations and homotopy colimits of simplicial sheaves, preprint (1998)

[19] R W Thomason, Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc. 85 (1979) 91 | DOI