Abstract
We extend the classical notion of a Reedy category so as to allow non-trivial automorphisms. Our extension includes many important examples occurring in topology such as Segal’s category Γ, or the total category of a crossed simplicial group such as Connes’ cyclic category Λ. For any generalized Reedy category \({\mathbb {R}}\) and any cofibrantly generated model category \({{\mathcal{E}}}\), the functor category \({{\mathcal{E}}^\mathbb {R}}\) is shown to carry a canonical model structure of Reedy type.
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References
Angeltveit V.: Enriched Reedy categories. Proc. Am. Math. Soc. 136, 2323–2332 (2008)
Antokoletz, E.R.: Nonabelian Algebraic Models for Classical Homotopy Types. PhD thesis, University of California at Berkeley (2008)
Barwick, C.: On Reedy Model Categories. arXiv:math/0708.2832
Batanin, M.A., Markl, M.: Crossed interval groups and operations on the Hochschild cohomology. arXiv:math/0803.2249
Berger C.: Iterated wreath product of the simplex category and iterated loop spaces. Adv. Math. 213, 230–270 (2007)
Bergner J.E.: Three models for the homotopy theory of homotopy theories. Topology 46, 397–436 (2007)
Borceux F.: Handbook of Categorical Algebra II. Enc Math, vol. 51. Cambridge University Press, Cambridge (1994)
Bousfield, A.K., Friedlander, E.M.: Homotopy theory of Γ-spaces, spectra, and bisimplicial sets. Lecture Notes in Math, vol. 658, pp. 80–130. Springer, Berlin (1978)
Bousfield A.K., Kan D.M.L.: Homotopy limits, completions, and localization. Lecture Notes in Mathematics, vol. 304. Springer, Berlin (1972)
Cisinski, D.-C.: Les préfaisceaux comme modèles des types d’homotopie. Astérisque, vol. 308. Soc. Math. France (2006)
Cisinski, D.-C., Moerdijk, I.: Dendroidal sets as a model for homotopy operads. arXiv:0809.3341
Cisinski, D.-C., Moerdijk, I.: Dendroidal Segal spaces and ∞-operads, 30 pp, preprint (2010)
Cisinski, D.-C., Moerdijk, I.: Dendroidal sets and simplicial operads (in preparation)
Connes A.: Cyclic homology and functor Ext n. C.R. Acad. Sci. Paris 296, 953–958 (1983)
Dwyer W.G., Hopkins M.J., Kan D.M.: The homotopy theory of cyclic sets. Trans. Am. Math. Soc. 291, 281–289 (1985)
Eilenberg S., Zilber J.A.: Semi-simplicial complexes and singular homology. Ann. Math. 51, 499–513 (1950)
Feigin, B.L., Tsygan B.L.: Additive K-theory. Lecture Notes in Mathematics, vol. 1289, pp. 67–209. Springer, Berlin (1987)
Fiedorowicz Z., Loday J.-L.: Crossed simplicial groups and their associated homology. Trans. Am. Math. Soc. 326, 57–87 (1991)
Gabriel, P., Zisman, M.: Calculus of fractions and homotopy theory. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 35. Springer, Berlin (1967)
Goerss P.F., Jardine R.J.: Simplicial homotopy theory. In: Progress in Mathematics, vol. 174, vi+510 pp. Birkhäuser, Basel (1999)
Haefliger A.: Extension of complexes of groups. Ann. Inst. Fourier 42, 275–311 (1992)
Hirschhorn, P.: Model Categories and Their Localizations. Math. Surveys Monogr., vol. 99. Am. Math. Soc. (2003)
Hovey, M.: Model Categories. Math. Surveys Monogr., vol. 63. Am. Math. Soc. (1999)
Joyal, A.: Disks, duality and θ-categories, 6 pp, preprint (1997)
Joyal A., Tierney M.: Quasi-categories vs Segal spaces. Contemp. Math. 431, 277–326 (2007)
Krasauskas R.: Skew-simplicial groups. Lithuanian Math. J. 27, 47–54 (1987)
Lurie, J.: Higher topos theory. In: Annals of Mathematics Studies, vol. 170, viii+925 pp. Princeton University Press, Princeton (2009)
Lydakis M.: Smash products and Γ-spaces. Math. Camb. Philos. Soc. 126, 311–328 (1999)
Moerdijk I.: Orbifolds as groupoids: an introduction. Contemp. Math. 310, 205–221 (2002)
Moerdijk I., Weiss I.: Dendroidal sets. Alg. Geom. Top. 7, 1441–1470 (2007)
Moerdijk I., Weiss I.: On inner Kan complexes in the category of dendroidal sets. Adv. Math. 221, 343–389 (2009)
Quillen, D.G.: Homotopical Algebra. Lecture Notes in Mathematics, vol. 43, iv+156 pp. Springer, Berlin (1967)
Quillen D.G.: Bivariant cyclic cohomology and models for cyclic homology types. J. Pure Appl. Algebra 101, 1–33 (1995)
Reedy, C.L.: Homotopy theories of model categories, 8 pp, preprint (1973)
Rezk C.: A model for the homotopy theory of homotopy theory. Trans. Am. Math. Soc. 353, 973–1007 (2003)
Segal G.: Categories and cohomology theories. Topology 13, 293–312 (1974)
Acknowledgements
The main results of this paper were presented at the CRM in Barcelona in February 2008, in the context of the Program on Homotopy Theory and Higher Categories. The actual writing of this paper was done while the second author was visiting the University of Nice in May 2008. He would like to express his gratitude to the CRM and the University of Nice for their hospitality and support. Both authors would like to thank the referee for his careful reading and useful suggestions.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Berger, C., Moerdijk, I. On an extension of the notion of Reedy category. Math. Z. 269, 977–1004 (2011). https://doi.org/10.1007/s00209-010-0770-x
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DOI: https://doi.org/10.1007/s00209-010-0770-x