# nLab model structure on cellular sets

model category

## Definitions

• category with weak equivalences

• weak factorization system

• homotopy

• small object argument

• resolution

• ## Universal constructions

• homotopy Kan extension

• Bousfield-Kan map

• ## Refinements

• monoidal model category

• enriched model category

• simplicial model category

• cofibrantly generated model category

• algebraic model category

• compactly generated model category

• proper model category

• stable model category

• ## Producing new model structures

• on functor categories (global)

• on overcategories

• Bousfield localization

• transferred model structure

• Grothendieck construction for model categories

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

• (∞,1)-category

• simplicial localization

• (∞,1)-categorical hom-space

• presentable (∞,1)-category

• ## Model structures

• Cisinski model structure
• ### for $\infty$-groupoids

for ∞-groupoids

• on topological spaces

• Strom model structure?
• Thomason model structure

• model structure on presheaves over a test category

• model structure on simplicial groupoids

• on cubical sets

• related by the Dold-Kan correspondence

• model structure on cosimplicial simplicial sets

• ### for $n$-groupoids

• for 1-groupoids

• ### for $\infty$-groups

• model structure on simplicial groups

• model structure on reduced simplicial sets

• ### for $\infty$-algebras

#### general

• on monoids

• on algebas over a monad

• on modules over an algebra over an operad

• #### specific

• model structure on differential-graded commutative algebras

• model structure on differential graded-commutative superalgebras

• on dg-algebras over an operad

• model structure on dg-modules

• ### for stable/spectrum objects

• model structure on spectra

• model structure on ring spectra

• model structure on presheaves of spectra

• ### for $(\infty,1)$-categories

• on categories with weak equivalences

• Joyal model for quasi-categories

• on sSet-categories

• for complete Segal spaces

• for Cartesian fibrations

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

• on dg-categories
• ### for $(\infty,1)$-operads

• on modules over an algebra over an operad

• ### for $(n,r)$-categories

• for (n,r)-categories as ∞-spaces

• for weak ∞-categories as weak complicial sets

• on cellular sets

• on higher categories in general

• on strict ∞-categories

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

• on homotopical presheaves

• model structure for (2,1)-sheaves/for 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).

## References

Last revised on October 8, 2013 at 20:35:15. See the history of this page for a list of all contributions to it.