Related papers: Complicial structures in the nerves of omega-categ…
We show that the nerve of a strict omega-category can be described algebraically as a simplicial set with additional operations subject to certain identities. The resulting structures are called sets with complicial identities. We also…
The nerve of a strict omega-category is a simplicial set with additional structure, making it into a so-called complicial set, and strict omega-categories are in fact equivalent to complicial sets. The nerve functor is represented by a…
This paper continues the development of a simplicial theory of weak omega-categories, by studying categories which are enriched in weak complicial sets. These complicial Gray-categories generalise both the Kan complex enriched categories of…
We show that the complicial nerve construction is homotopically compatible with two flavors of cone constructions when starting with an $\omega$-category that is suitably free and loop-free. An instance of the result recovers the fact that…
It is shown that the cubical nerve of a strict omega-category is a sequence of sets with cubical face operations and distinguished subclasses of thin elements satisfying certain thin filler conditions. It is also shown that a sequence of…
The aim of these notes is to introduce the intuition motivating the notion of a "complicial set", a simplicial set with certain marked "thin" simplices that witness a composition relation between the simplices on their boundary. By varying…
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…
It has been conjectured since the 1980s that Verity's $n$-complicial sets were a model for $(\infty,n)$-categories. This text is dedicated to providing a positive answer to this conjecture. The proof of this result relies on a thorough…
Many monoidal-type objects are known to be classified by maps from the Catalan simplicial set $\mathbb{C}$ to various nerves of categories and higher categories. There are, for example, three different nerves of the 2-category of categories…
Our aim is to compare three nerve functors for strict $n$-categories: the Street nerve, the cellular nerve and the multi-simplicial nerve. We show that these three functors are equivalent in some appropriate sense. In particular, the…
We show that the homotopy theory of strict 2-categories embeds in that of $(\infty,2)$-categories in the form of 2-precomplicial sets. More precisely, we construct a nerve-categorification adjunction that is a Quillen pair between Lack's…
We show that the classification diagram of a relative $\infty$-category arising from a relative simplicial category is equivalent to the levelwise nerve. Applications include the comparison of the diagonal of the levelwise nerve and the…
There are several ways to construct omega-categories from combinatorial objects such as pasting schemes or parity complexes. We make these constructions into a functor on a category of chain complexes with additional structure, which we…
We establish a Quillen equivalence relating the homotopy theory of Segal operads and the homotopy theory of simplicial operads, from which we deduce that the homotopy coherent nerve functor is a right Quillen equivalence from the model…
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…
The paper focuses on investigating how certain relations between strict $n$-categories are preserved in a particular implementation of $(\infty,n)$-categories, given by saturated $n$-complicial sets. In this model, we show that the…
The most natural notion of a simplicial nerve for a (weak) bicategory was given by Duskin, who showed that a simplicial set is isomorphic to the nerve of a $(2,1)$-category (i.e. a bicategory with invertible $2$-morphisms) if and only if it…
This paper develops the foundations of a simplicial theory of weak omega-categories, which builds upon the insights originally expounded by Ross Street in his 1987 paper on oriented simplices. The resulting theory of weak complicial sets…
In this dissertation, we compare the "classical" homology of an $\omega$-category (defined as the homology of its Street nerve) with its polygraphic homology. More precisely, we prove that both homologies generally do not coincide and call…
The notion of geometric nerve of a 2-category (Street, \cite{refstreet}) provides a full and faithful functor if regarded as defined on the category of 2-categories and lax 2-functors. Furthermore, lax 2-natural transformations between lax…