A Cartesian presentation of weak \(n\)-categories. (English) Zbl 1203.18015

Geom. Topol. 14, No. 1, 521-571 (2010); correction ibid. 14, No. 4, 2301-2304 (2010).
In a previous paper, the author developed a model for the homotopy theory of homotopy theories, namely that of complete Segal spaces. These simplicial spaces satisfied some conditions allowing them to be regarded as up-to-homotopy versions of simplicial categories. Furthermore, they are the fibrant objects in a cartesian model structure on the category of simplicial spaces which is obtained as a localization of the Reedy structure. This work has become of particular interest because complete Segal spaces can be thought of as \((\infty, 1)\)-categories, or \(\infty\)-categories with all morphisms invertible after level one.
This current paper is an extension of this work, in that it provides a model for more general \((\infty, n)\)-categories, where morphisms are now invertible above dimension \(n\). The idea is to replace simplicial diagrams of simplicial sets by diagrams indexed by an inductively defined category \(\Theta_n\) whose objects can be thought of as certain basic kinds of strict \(n\)-categories. As before, there is a cartesian model structure on the category of all such diagrams, and the fibrant objects satisfy higher-order variations of the conditions for complete Segal spaces. In particular, a Segal condition guarantees that composition is defined up to homotopy at all levels, and a completeness condition says that the simplicial set associated to a single \(k\)-morphism is a kind of moduli space which is equivalent to the space of homotopy equivalences in the simplicial set associated to a single \((k+1)\)-morphism.
Different characterizations of these fibrant objects are given, and the special cases of groupoid objects (where all morphisms are weakly invertible) and truncated objects modeling \((m,n)\)-categories are also considered. In particular, groupoid objects in the truncated setting are shown to satisfy the homotopy hypothesis, a condition generally expected of any model for higher categories.
It should be noted that a correction to the proof of Proposition 6.6 has been published [see C. Rezk, Geom. Topol. 14, No. 4, 2301–2304 (2010; Zbl 1203.18016)].


18G55 Nonabelian homotopical algebra (MSC2010)
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
55U40 Topological categories, foundations of homotopy theory
18G30 Simplicial sets; simplicial objects in a category (MSC2010)


Zbl 1203.18016
Full Text: DOI arXiv


[1] J C Baez, J Dolan, Categorification (editors E Getzler, M Kapranov), Contemp. Math. 230, Amer. Math. Soc. (1998) 1 · Zbl 0923.18002
[2] C Barwick, Extended research statement (2007)
[3] C Berger, A cellular nerve for higher categories, Adv. Math. 169 (2002) 118 · Zbl 1024.18004
[4] C Berger, Iterated wreath product of the simplex category and iterated loop spaces, Adv. Math. 213 (2007) 230 · Zbl 1127.18008
[5] J E Bergner, A model category structure on the category of simplicial categories, Trans. Amer. Math. Soc. 359 (2007) 2043 · Zbl 1114.18006
[6] J E Bergner, Three models for the homotopy theory of homotopy theories, Topology 46 (2007) 397 · Zbl 1119.55010
[7] E Cheng, Higher dimensional categories: An illustrated guide book (2004)
[8] D Dugger, Universal homotopy theories, Adv. Math. 164 (2001) 144 · Zbl 1009.55011
[9] P S Hirschhorn, Model categories and their localizations, Math. Surveys and Monogr. 99, Amer. Math. Soc. (2003) · Zbl 1017.55001
[10] A Hischowitz, C Simpson, Descente pour les \(n\)-champs
[11] M Hovey, Model categories, Math. Surveys and Monogr. 63, Amer. Math. Soc. (1999) · Zbl 0909.55001
[12] A Joyal, Disks, duality, and \(\Theta\)-categories, Preprint (1999)
[13] A Joyal, M Tierney, Quasi-categories vs Segal spaces (editors A Davydov, M Batanin, M Johnson, S Lack, A Neeman), Contemp. Math. 431, Amer. Math. Soc. (2007) 277 · Zbl 1138.55016
[14] T Leinster, A survey of definitions of \(n\)-category, Theory Appl. Categ. 10 (2002) 1 · Zbl 0987.18007
[15] J Lurie, On the classification of topological field theories · Zbl 1180.81122
[16] C Rezk, A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353 (2001) 973 · Zbl 0961.18008
[17] Z Tamsamani, Sur des notions de \(n\)-catégorie et \(n\)-groupoïde non strictes via des ensembles multi-simpliciaux, \(K\)-Theory 16 (1999) 51 · Zbl 0934.18008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.