Complicial sets, an overture

TL;DR

This paper introduces complicial sets, a framework for modeling weak higher categories using marked simplicial sets, and explores their homotopy theory and relation to strict $ ext{omega}$-categories.

Contribution

It develops the theory of complicial sets, including their definitions, model structures, and connections to strict $ ext{omega}$-categories, advancing the understanding of weak infinite-dimensional categories.

Findings

Complicial sets can model $( abla,n)$-categories for all $n \\geq 0$ including infinity.

The Street nerve embeds strict $ ext{omega}$-categories into complicial sets.

A saturation condition on marked simplices leads to various homotopy-theoretic model structures.

Abstract

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 categories; the second defining the Street nerve, which embeds strict $\omega$-categories as "strict" complicial sets; and the third exploring an important saturation condition on the marked simplices in a complicial set and presenting a variety of model structures that capture their basic homotopy theory. Scattered throughout are suggested exercises for the reader who wants to engage more deeply with these notions.