Higher quasi-categories vs higher Rezk spaces

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 Θ_n. 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 Θ_n-spaces showing that n-quasi-categories are a model for (\infty, n)-categories. For n = 1, we recover the two Quillen equivalences defined by Joyal and Tierney between quasi-categories and complete Segal spaces.

44 pages, v2: terminology changed (see Remark 5.27), Corollary 7.5 added, appendix A added, references added, v3: reorganization of Sections 5 and 6, more informal comments, new section characterizing strict n-categories whose nerve is an n-quasi-category, numbering has changed

References in corpus (3)

• Comparison of models for $(\infty ,n)$-categories, I
• On the homotopy theory of Grothendieck \infty-groupoids
• Strict \infty-groupoids are Grothendieck \infty-groupoids

Cited by in corpus (15)

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• The Gray tensor product for 2-quasi-categories
• Quasi-Categories vs. Segal Spaces: Cartesian Edition
• An $(\infty ,2)$-categorical pasting theorem
• Categories of graphs for operadic structures
• On autoequivalences of the (\infty, 1)-category of \infty-operads
• Inner horns for 2-quasi-categories
• Quasi-2-Segal sets
• Homotopy theories of $(\infty ,\infty )$-categories as universal fixed points with respect to enrichment
• Joyal's cylinder conjecture
• A cubical model for $(\infty ,n)$-categories
• Discreteness and completeness for ${\Theta }_n$-models of $(\infty ,n)$-categories