# nLab model structure on cellular sets

Contents

### Context

#### Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of $(\infty,1)$-categories

Model structures

for $\infty$-groupoids

for ∞-groupoids

for equivariant $\infty$-groupoids

for rational $\infty$-groupoids

for rational equivariant $\infty$-groupoids

for $n$-groupoids

for $\infty$-groups

for $\infty$-algebras

general $\infty$-algebras

specific $\infty$-algebras

for stable/spectrum objects

for $(\infty,1)$-categories

for stable $(\infty,1)$-categories

for $(\infty,1)$-operads

for $(n,r)$-categories

for $(\infty,1)$-sheaves / $\infty$-stacks

#### Higher category theory

higher category theory

# Contents

## Idea

For $\Theta_n$ the $n$-truncation of the Theta-category, an $n$-cellular set, hence a presheaf on $\Theta_n$, may be viewed as a collection of n-morphisms of an “$n$-graph” underlying an (∞,n)-category. The model structure on cellular sets (Ara) models $(\infty,n)$-categories this way. This model is referred to as n-quasicategories.

## Definition

Let $\Theta_n$ be the Theta category restricted to $n$-cells. The model structure on cellular sets is the Cisinski model structure on the category of presheaves $PSh(\Theta_n)$ defined by the following localizer: (…)

In (Ara) the fibrant objects in this model structure are called $n$-quasi-categories, see Relation to quasi-categories below.

## Properties

### Relation to quasi-categories

For $n = 1$ we have $\Theta_1 \simeq \Delta$ is the simplex category; and the model structure on 1-cellular sets reproduces the model structure for quasi-categories. (Ara, theorem 5.26)

### Relation to $\Theta_n$-spaces

The model structure on $\Theta_n$-cellular sets is Quillen equivalent to that ∞-n spaces. (Ara, theorem 7.4).