Document Zbl 1291.18009 - zbMATH Open

The most basic notion of a homotopy theory is a category equipped with some notion of weak equivalences. Work of Dwyer and Kan established that this data is equivalent to that of a simplicial or topological category, and more recently, these structures were in turn shown to be equivalent to other models such as quasicategories and complete Segal spaces, via Quillen equivalences of model categories. In a previous paper, the authors showed that categories with weak equivalences, which they call relative categories, themselves are objects of a model category which is equivalent to the others. All these different structures can be considered as ways to think about homotopy theories, but also as \((\infty,1)\)-categories. However, one can ask to generalize each of these structures to get models for more general \((\infty, n)\)-categories. For example, complete Segal spaces can be generalized in two ways: to the \(n\)-fold Segal spaces of Barwick and Lurie, and to the \(\Theta_n\)-spaces of Rezk. In this paper, the authors generalize their definition of relative category to that of an \(n\)-relative category. Roughly speaking, this structure consists of a category, together with a choice of \(n\) different subcategories which all share a subcategory of weak equivalences. An adjoint pair of functors relates \(n\)-relative categories to \(n\)-fold complete Segal spaces, generalizing the authors’ previous work and establishing \(n\)-relative categories as a model for \((\infty, n)\)-categories.

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