Minimal fibrations of dendroidal sets

Moerdijk, Ieke¹ ; Nuiten, Joost²

¹Mathematical Institute, Utrecht University, PO Box 80010, 3508 TA Utrecht, Netherlands
²Mathematical Institute, Utrecht University, P.O. Box 80010, 3508 TA Utrecht, Netherlands

Algebraic and Geometric Topology, Tome 16 (2016) no. 6, pp. 3581-3614

Résumé

We prove the existence of minimal models for fibrations between dendroidal sets in the model structure for ∞–operads, as well as in the covariant model structure for algebras and in the stable one for connective spectra. We also explain how our arguments can be used to extend the results of Cisinski (2014) and give the existence of minimal fibrations in model categories of presheaves over generalized Reedy categories of a rather common type. Besides some applications to the theory of algebras over ∞–operads, we also prove a gluing result for parametrized connective spectra (or Γ–spaces).

Détail

DOI : 10.2140/agt.2016.16.3581

Classification : 55R65, 55U35, 55P48, 18D50
Keywords: minimal fibrations, dendroidal sets, Gamma-spaces, Reedy categories

Bibliographie

[1] M G Barratt, V K A M Gugenheim, J C Moore, On semisimplicial fibre-bundles, Amer. J. Math. 81 (1959) 639 | DOI

[2] C Barwick, On left and right model categories and left and right Bousfield localizations, Homology, Homotopy Appl. 12 (2010) 245 | DOI

[3] M Bašić, T Nikolaus, Dendroidal sets as models for connective spectra, J. K-Theory 14 (2014) 387 | DOI

[4] B Van Den Berg, I Moerdijk, W-types in homotopy type theory, Math. Structures Comput. Sci. 25 (2015) 1100 | DOI

[5] C Berger, I Moerdijk, On an extension of the notion of Reedy category, Math. Z. 269 (2011) 977 | DOI

[6] D C Cisinski, Les préfaisceaux comme modèles des types d’homotopie, 308, Société Mathématique de France (2006)

[7] D C Cisinski, Univalent universes for elegant models of homotopy types, preprint (2014)

[8] D C Cisinski, I Moerdijk, Dendroidal sets as models for homotopy operads, J. Topol. 4 (2011) 257 | DOI

[9] D C Cisinski, I Moerdijk, Dendroidal sets and simplicial operads, J. Topol. 6 (2013) 705 | DOI

[10] D C Cisinski, I Moerdijk, Note on the tensor product of dendroidal sets, preprint (2014)

[11] A Connes, Cohomologie cyclique et foncteurs Extn, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983) 953

[12] W G Dwyer, M J Hopkins, D M Kan, The homotopy theory of cyclic sets, Trans. Amer. Math. Soc. 291 (1985) 281 | DOI

[13] P Gabriel, M Zisman, Calculus of fractions and homotopy theory, 35, Springer (1967) | DOI

[14] G Heuts, Algebras over infinity-operads, preprint (2011)

[15] A Joyal, The theory of quasi-categories and its applications, preprint (2008)

[16] J Lurie, Higher topos theory, 170, Princeton University Press (2009) | DOI

[17] J P May, Simplicial objects in algebraic topology, 11, D. Van Nostrand Co. (1967)

[18] I Moerdijk, Lectures on dendroidal sets, from: "Simplicial methods for operads and algebraic geometry" (editors C Casacuberta, J Kock), Springer (2010) 1 | DOI

[19] I Moerdijk, I Weiss, Dendroidal sets, Algebr. Geom. Topol. 7 (2007) 1441 | DOI

[20] I Moerdijk, I Weiss, On inner Kan complexes in the category of dendroidal sets, Adv. Math. 221 (2009) 343 | DOI

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

[22] D G Quillen, The geometric realization of a Kan fibration is a Serre fibration, Proc. Amer. Math. Soc. 19 (1968) 1499 | DOI

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

[24] J Rosický, W Tholen, Left-determined model categories and universal homotopy theories, Trans. Amer. Math. Soc. 355 (2003) 3611 | DOI

[25] G Segal, Categories and cohomology theories, Topology 13 (1974) 293 | DOI