nLab
model structure on cellular sets

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).

References

• Dimitri Ara, Higher quasi-categories vs higher Rezk spaces (arXiv:1206.4354)