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)].

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)

Zbl 1203.18016
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References:

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