##
**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)].

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)].

Reviewer: Julia Bergner (Riverside)

### MSC:

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) |

### Keywords:

\((\infty, n)\)-categories; complete Segal spaces; Reedy structure; strict \(n\)-categories### Citations:

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