Related papers: An explicit comparison between $2$-complicial sets…

In this paper we complete a chain of explicit Quillen equivalences between the model category for $\Theta_{n+1}$-spaces and the model category of small categories enriched in $\Theta_n$-spaces. The Quillen equivalences given here connect…

This thesis is divided into two parts. In the first part, we study models of $(\infty,\omega)$-categories. The main result is to establish a Quillen equivalence between Rezk's complete Segal $\Theta$-spaces and Verity's complicial sets. In…

We prove the compatibility between the suspension construction and the complicial nerve of $\omega$-categories. As a motivating application, we produce a Quillen pair between the models of $(\infty,n)$-categories given by Rezk's complete…

We establish cartesian model structures for variants of $\Theta_n$-spaces in which we replace some or all of the completeness conditions by discreteness conditions. We prove that they are all equivalent to each other and to the…

We show that the 2-Segal spaces (also called decomposition spaces) of Dyckerhoff-Kapranov and G\'alvez-Kock-Tonks have a natural analogue within simplicial sets, which we call quasi-2-Segal sets, and that the two ideas enjoy a similar…

We show that complete Segal spaces and Segal categories are Quillen equivalent to quasi-categories.

We give a new proof of the equivalence between two of the main models for $(\infty,n)$-categories, namely the $n$-fold Segal spaces of Barwick and the $\Theta_{n}$-spaces of Rezk, by proving that these are algebras for the same monad on the…

We lift Charles Rezk's complete Segal space model structure on the category of simplicial spaces to a Quillen equivalent one on the category of relative categories.

We establish Rezk completion functors for $\Theta_n$-spaces with respect to each and all of the completeness conditions. As a consequence, we obtain a characterization completeness of Segal $\Theta_n$-spaces as locality with respect to…

In previous work, we showed that there are appropriate model category structures on the category of simplicial categories and on the category of Segal precategories, and that they are Quillen equivalent to one another and to Rezk's complete…

We introduce a notion of n-quasi-categories as fibrant objects of a model category structure on presheaves on Joyal's n-cell category \Theta_n. Our definition comes from an idea of Cisinski and Joyal. However, we show that this idea has to…

In a previous paper we lifted Charles Rezk's complete Segal model structure on the category of simplicial spaces to a Quillen equivalent one on the category of "relative categories," and our aim in this successor paper is to obtain a more…

We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: On marked simplicial sets, on bisimplicial spaces, on bisimplicial sets, on marked simplicial spaces. The main…

In this paper, we construct a model structure for $(\infty,1)$-categories on the category of simplicial spaces, whose fibrant objects are the Segal spaces. In particular, we show that it is Quillen equivalent to the models of…

We axiomatise the theory of $(\infty,n)$-categories. We prove that the space of theories of $(\infty,n)$-categories is a $B(\mathbb{Z}/2)^n$. We prove that Rezk's complete Segal $\Theta_n$-spaces, Simpson and Tamsamani's Segal…

We put a model structure on the category of categories internal to simplicial sets whose weak equivalences are reflected by the nerve functor to bisimplicial sets with Rezk's model structure. This model structure is shown to be Quillen…

We show that the Rezk classification diagram of a relative category admitting a homotopical version of the two-sided calculus of fractions is a Segal space up to Reedy-fibrant replacement. This generalizes the result of Rezk and Bergner on…

We introduce rational $(\infty, 1)$-categories, which are $(\infty, 1)$-categories enriched in spaces whose higher homotopy groups are rational vector spaces. We provide two models for rational $(\infty, 1)$-categories, rational complete…

We introduce a relative version of the $2$-Segal simplicial spaces defined by Dyckerhoff and Kapranov and G\'{a}lvez-Carrillo, Kock and Tonks. Examples of relative $2$-Segal spaces include the categorified unoriented cyclic nerve, real…

We introduce the dendroidal analogs of the notions of complete Segal space and of Segal category, and construct two appropriate model categories for which each of these notions corresponds to the property of being fibrant. We prove that…