Dominic Verity
The primary purpose of this work is to characterise strict ω-categories as simplicial sets with structure. We prove the Street-Roberts conjecture which states that they are exactly the ``complicial sets'' defined and named by John Roberts in his handwritten notes of that title.
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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 homotopy theory and the Gray-categories of weak 3-category theory. We derive a simplicial nerve construction, which is closely related to Cordier and Porter's homotopy coherent nerve, and show that this faithfully represents complicial Gray-categories as weak complicial sets. The category of weak complicial sets may itself be canonically enriched to a compl…
We show that the pretensor and tensor products of simplicial sets with marking are compatible with the homotopy theory of saturated $N$-complicial sets (which are a proposed model of $(\infty,N)$-categories), in the form of a Quillen bifunctor and a homotopical bifunctor, respectively.
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 $(\infty,n)$-categorical nerve of $n$-categories is homotopically compatible with $1$-categorical suspension and wedge. As an application, we show that certain pushouts encoding composition in $n$-categories are homotopy pushouts of saturated $n$-complicial sets.
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 the marking conventions, complicial sets can be used to model $(\infty,n)$-categories for each $n \geq 0$, including $n=\infty$. For this reason, complicial sets present a fertile setting for thinking about weak infinite dimensional categories in varying dimensions. This overture is presented in three acts: the first introducing simplicial models of higher c…
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 provides a common generalisation of the theories of (strict) omega-categories, Kan complexes and Joyal's quasi-categories. We generalise a number of results due to the current author with regard to complicial sets and strict omega-categories to provide an armoury of well behaved technical devices, such as joins and Gray tensor products, which will be used to s…
It is known that strict omega-categories are equivalent through the nerve functor to complicial sets and to sets with complicial identities. It follows that complicial sets are equivalent to sets with complicial identities. We discuss these equivalences. In particular we give a conceptual proof that the nerves of omega-categories are complicial sets, and a direct proof that complicial sets are sets with complicial identities.
We prove the compatibility between the suspension construction and the complicial nerve of $ω$-categories. As a motivating application, we produce a Quillen pair between the models of $(\infty,n)$-categories given by Rezk's complete Segal $Θ_n$-spaces and Verity's $n$-complicial sets.
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 sequence of strict omega-categories, called orientals, which are associated to simplexes. In this paper we give a detailed algebraic description of the morphisms between orientals. The aim is to describe complicial sets algebraically, by operators and equational axioms.
The complicial model of $(\infty,ω)$-categories
研究$(\infty,ω)$-范畴的复形模型，核心方法是证明Verity n-复形集性质，贡献是给出相关模型。
In this note, we study the connection between Gray tensor product and suspension. We derive a characterization of weak equivalences as fully faithful and essentially surjective functors. We construct the $co$ duality, a weak involution that reverses the direction of even dimensional cells. We conclude by studying Grothendieck fibrations of complicial sets.
A cubical model for $(\infty, n)$-categories
本文提出了一种新的$(\infty,n)$-范畴模型，基于标记立方集，并展示了其在Gray张量积下的单向性和三角化 functor 的左Quillen性质。
We prove the existence of a model structure on the category of stratified simplicial sets whose fibrant objects are precisely $n$-complicial sets, which are a proposed model for $(\infty,n)$-categories, based on previous work of Verity and Riehl. We then construct a Quillen equivalent model based on simplicial presheaves over a category that can facilitate the comparison with other established models.
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 model structure for 2-categories and Riehl-Verity's model structure for 2-complicial sets. Furthermore, we show that Lack's model structure is transferred along this nerve and that the nerve is homotopically fully faithful.
Equivalence of cubical and simplicial approaches to $(\infty,n)$-categories
本文证明了立方体与单纯形方法在$(\infty,n)$-范畴中的等价性，基于锥形理论的理论发展。
We prove that each of the model structures for ($n$-trivial, saturated) comical sets on the category of marked cubical sets having only faces and degeneracies (without connections) is Quillen equivalent to the corresponding model structure for ($n$-trivial, saturated) complicial sets on the category of marked simplicial sets, as well as to the corresponding comical model structures on cubical sets with connections. As a consequence, we show that the cubical Joyal model structure on cubical sets without connections is equivalent to its analogues on cubical sets with connections and to the Joyal…
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 construct an equivalence between the categories of strict omega-categories and of sets with complical identities.
The orientals are the free strict $ω$-categories on the simplices introduced by Street. The aim of this paper is to show that they are also the free weak $ω$-categories on the same generating data. More precisely, we exhibit the Street nerves of the orientals as fibrant replacements of the simplices in Verity's model structure for complicial sets.
An inductive model structure for strict $\infty$-categories
本文为标记严格$\infty$-范畴构建了左半模型结构，通过Street神经建立与复集的Quillen伴随，并据此证明了弱$(\infty,\infty)$-范畴存在至少三种不同的非等价概念。
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