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Higher Quasi-Categories vs Higher Rezk Spaces

Published online by Cambridge University Press:  22 May 2015

Dimitri Ara
Dimitri Ara*
Affiliation:
Institut de Mathématiques de Jussieu, Université Paris Diderot – Paris 7, Case 7012, Bâtiment Chevaleret, 75205 Paris Cedex 13, France, E-mail address: ara@math.jussieu.fr, URL: http://people.math.jussieu.fr/~ara/
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Abstract

We introduce a notion of n-quasi-categories as fibrant objects of a model category structure on presheaves on Joyal's n-cell category Θn. Our definition comes from an idea of Cisinski and Joyal. However, we show that this idea has to be slightly modified to get a reasonable notion. We construct two Quillen equivalences between the model category of n-quasi-categories and the model category of Rezk Θn-spaces, showing that n-quasi-categories are a model for (∞, n)-categories. For n = 1, we recover the two Quillen equivalences defined by Joyal and Tierney between quasi-categories and complete Segal spaces.

Keywords

18D0518D2018E3518G3018G5555U1055U3555U40∞, n-categoriesquasi-categoriesn-quasi-categoriesSegal spacesΘn-spacesA-localizers
Type
Research Article
Information
Journal of K-Theory , Volume 14 , Issue 3 , December 2014 , pp. 701 - 749
Copyright
Copyright © ISOPP 2014 

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References

1.Ara, D., Sur les ∞-groupoïdes de Grothendieck et une variante ∞-catégorique, Ph.D. thesis, Université Paris Diderot – Paris 7, Supervised by Maltsiniotis, G., 2010.Google Scholar
2.Ara, D., The groupoidal analogue Θ̃ to Joyal’s category Θ is a test category, Appl. Categor. Structures, 2012.Google Scholar
3.Ara, D., On the homotopy theory of Grothendieck ∞-groupoids, J. Pure Appl. Algebra 217 (2013), no. 7, 1237–1278.CrossRefGoogle Scholar
4.Ara, D., Strict ∞-groupoids are Grothendieck ∞-groupoids, J. Pure Appl. Algebra 217 (2013), no. 12, 2298–2312.Google Scholar
5.Ara, D. and Métayer, F., The Brown-Golasmski model structure on strict ∞-groupoids revisited, Homology Homotopy Appl. 13 (2011), no. 1, 121–142.CrossRefGoogle Scholar
6.Barwick, C. and Kan, D.M., Relative categories: another model for the homotopy theory of homotopy theories, Indag. Math. (M.S.) 23 (2012), no. 1–2, 42–68.Google Scholar
7.Barwick, C. and Kan, D.M., n-relative categories: a model for the homotopy theory of n-fold homotopy theories, Homology Homotopy Appl. 15 (2013), no. 2, 281–300.Google Scholar
8.Barwick, C. and Schommer-Pries, C., On the unicity of the homotopy theory of higher categories, arXiv:1112.0040v4 [math.AT], 2013.Google Scholar
9.Batanin, M.A., Monoidal globular categories as a natural environment for the theory of weak n-categories, Adv. Math. 136 (1998), no. 1, 39–103.CrossRefGoogle Scholar
10.Bénabou, J., Introduction to bicategories, Reports of the Midwest Category Seminar, Springer-Verlag, 1967, pp. 1–77.Google Scholar
11.Berger, C., A cellular nerve for higher categories, Adv. Math. 169 (2002), no. 1, 118–175.Google Scholar
12.Berger, C., Iterated wreath product of the simplex category and iterated loop spaces, Adv. Math. 213 (2007), no. 1, 230–270.Google Scholar
13.Bergner, J.E., A model category structure on the category of simplicial categories, Trans. Amer. Math. Soc. 359 (2007), no. 5, 2043–2058.CrossRefGoogle Scholar
14.Bergner, J.E., Three models for the homotopy theory of homotopy theories, Topology 46 (2007), no. 4, 397–436.Google Scholar
15.Bergner, J.E., A survey of (∞, 1)-categories, Towards higher categories, IMA Vol. Math. Appl., vol. 152, Springer-Verlag, 2010, pp. 69–83.Google Scholar
16.Bergner, J.E., Models for (∞, n)-categories and the cobordism hypothesis, Mathematical foundations of quantum field theory and perturbative string theory, Proc. Sympos. Pure Math., vol. 83, Amer. Math. Soc., 2011, pp. 17–30.Google Scholar
17.Bergner, J.E. and Rezk, C., Comparison of models for (∞, n)-categories, I, Geom. Topol. 17 (2013), no. 4, 2163–2202.Google Scholar
18.Bergner, J.E. and Rezk, C., Reedy categories andthe Θ-construction, Math. Z. 274 (2013), no. 1–2, 499–514.Google Scholar
19.Boardman, J.M. and Vogt, R.M., Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics, Vol. 347, Springer-Verlag, 1973.CrossRefGoogle Scholar
20.Cisinski, D.-C., Les préfaisceaux comme modèles des types d’homotopie, Astérisque, no. 308, Soc. Math. France, 2006.Google Scholar
21.Dugger, D., Replacing model categories with simplicial ones, Trans. Amer. Math. Soc. 353 (2001), no. 12, 5003–5027.Google Scholar
22.Gabriel, P. and Zisman, M., Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 35, Springer-Verlag, 1967.Google Scholar
23.Gindi, H., A homotopy theory of weak ω-categories, arXiv:1207.0860v2 [math.CT], 2012.Google Scholar
24.Grothendieck, A., Pursuing stacks, Manuscript, 1983, to be published in Documents Mathématiques.Google Scholar
25.Hirschhorn, P., Model categories and their localizations, Mathematical Surveys and Monographs, vol. 99, American Mathematical Society, 2003.Google Scholar
26.Hirschowitz, A. and Simpson, C., Descente pour les n-champs, arXiv:math/9807049v3 [math.AG], 2001.Google Scholar
27.Joyal, A., Disks, duality and Θ-categories, Preprint, 1997.Google Scholar
28.Joyal, A., Quasi-categories and Kan complexes, J. Pure Appl. Algebra 175 (2002), no. 1–3, 207–222.Google Scholar
29.Joyal, A., Quasi-categories vs simplicial categories, Preprint, 2007.Google Scholar
30.Joyal, A., Notes on quasi-categories, Preprint, 2008.Google Scholar
31.Joyal, A., The theory of quasi-categories and its applications, Preprint, 2008.Google Scholar
32.Joyal, A. and Tierney, M., Quasi-categories vs Segal spaces, Categories in algebra, geometry and mathematical physics, Contemp. Math., vol. 431, Amer. Math. Soc., 2007, pp. 277–326.Google Scholar
33.Leinster, T., A survey of definitions of n-category, Theory Appl. Categ. 10 (2002), 1–70.Google Scholar
34.Leinster, T., Higher operads, higher categories, London Mathematical Society Lecture Note Series, vol. 298, Cambridge University Press, 2004.Google Scholar
35.Lurie, J., Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, 2009.Google Scholar
36.Lurie, J., Higher algebra, Preprint, 2014.Google Scholar
37.Makkai, M. and Zawadowski, M., Duality for simple ω-categories and disks, Theory Appl. Categ. 8 (2001), 114–243.Google Scholar
38.Maltsiniotis, G., Infini groupoïdes non stricts, d’après Grothendieck, Preprint, 2007.Google Scholar
39.Maltsiniotis, G., Infini catégories non strictes, une nouvelle définition, Preprint, 2007.Google Scholar
40.Maltsiniotis, G., Grothendieck ∞-groupoids and still another definition of ∞-categories, arXiv:1009.2331v1 [math.CT], 2010.Google Scholar
41.Quillen, D.G., Homotopical algebra, Lecture Notes in Mathematics, vol. 43, Springer-Verlag, 1967.Google Scholar
42.Rezk, C., A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353 (2001), no. 3, 973–1007.CrossRefGoogle Scholar
43.Rezk, C., A Cartesian presentation of weak n-categories, Geom. Topol. 14 (2010), no. 1, 521–571.Google Scholar
44.Rezk, C., Correction to “A Cartesian presentation of weak n-categories”, Geom. Topol. 14 (2010), no. 4, 2301–2304.CrossRefGoogle Scholar
45.Simpson, C., Homotopy theory of higher categories, New mathematical monographs, vol. 19, Cambridge University Press, 2011.Google Scholar
46.Weber, M., Familial 2-functors and parametric right adjoints, Theory Appl. Categ. 18 (2007), 665–732.Google Scholar