nLab
model structure on cellular sets

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

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

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

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

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

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Idea

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

Definition

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

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

Properties

Relation to quasi-categories

For n=1n = 1 we have Θ 1Δ\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 Θ n\Theta_n-spaces

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

References

Revised on October 8, 2013 20:35:15 by Urs Schreiber (89.204.137.109)