Weak complicial sets, a simplicial weak omega-category theory. Part II: nerves of complicial Gray-categories

Nerves of 2-categories and 2-categorification of $(\infty,2)$-categories

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.

Complicial sets, an overture

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…

Dualities in the complicial model of $\infty$-categories

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.

Weak complicial sets, a simplicial weak omega-category theory. Part I: basic homotopy theory

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…

The algebra of the nerves of omega-categories

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.

一个$(\infty,n)$-范畴的立方模型

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。

关于行走余归纳等价的可收缩性

About the contractibility of the walking coinductive equivalence

研究行走余归纳等价的Roberts-Street神经得到的带标记单纯集，证明其为不可收缩的饱和复形集且有限截断均可收缩，给出右(∞,∞)-范畴不可收缩但到左(∞,∞)-范畴的反射可收缩的明确例子

Orientals as free weak $ω$-categories

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.

Conditions de Kan sur les nerfs des $ω$-catégories

We show that the Street nerve of a strict $ω$-category $C$ is a Kan complex (respectively a quasi-category) if and only if the $n$-cells of $C$ for $n\geq 1$ (respectively $n> 1$) are weakly invertible. Moreover, we equip $\mathcal{N}(C)$ with a structure of saturated complicial set where the $n$-simplices correspond to morphisms from the $n^{th}$ oriental to $C$ sending the unique non-trivial $n$-cell of the domain to a weakly invertible cell of $C$.

$(\infty,\omega)$-范畴的理论与模型

Theory and models of $(\infty,ω)$-categories

本文建立$(\infty,\omega)$-范畴的两种模型（完备Segal $\Theta$-空间与complicial集合）的Quillen等价，并利用其联系在$(\infty,\omega)$-范畴中实现标准范畴构造，重点研究Grothendieck构造。

严格$\infty$-范畴的一个归纳模型结构

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。

更严格$n$-范畴的同伦理论

Homotopy theory of stricter $n$-categories

本研究基于正则有向复形理论定义更严格$n$-范畴，推导其Gray积与纬垂稳定性等性质，构建folk模型结构并证明其与图解集模型结构的右转移关系，丰富了高维范畴同伦理论工具。

2-Dimensional Categories

This book is an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories, pasting diagrams, lax functors, 2-/bilimits, the Duskin nerve, 2-nerve, adjunctions and monads in bicategories, 2-monads, biequivalences, the Bicategorical Yoneda Lemma, and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double cate…