nLab
model structure on cellular sets
Contents
Context
Model category theory
model category , model $\infty$ -category

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
specific
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

Basic concepts
Basic theorems
Applications
Models
Morphisms
Functors
Universal constructions
Extra properties and structure
1-categorical presentations
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 ).

References
Last revised on February 9, 2021 at 09:08:15.
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