nLab
model structure on cellular sets
Context
Model category theory
model category

Definitions
category with weak equivalences

weak factorization system

homotopy

small object argument

resolution

Morphisms
Quillen adjunction

Universal constructions
homotopy Kan extension

homotopy limit /homotopy colimit

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

cartesian closed model category , locally cartesian closed 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

Thomason model structure

model structure on presheaves over a test category

on simplicial sets , on semi-simplicial sets

model structure on simplicial groupoids

on cubical sets

on strict ∞-groupoids , on groupoids

on chain complexes /model structure on cosimplicial abelian groups

related by the Dold-Kan correspondence

model structure on cosimplicial simplicial sets

for $n$ -groupoids
for n-groupoids /for n-types

for 1-groupoids

for $\infty$ -groups
model structure on simplicial groups

model structure on reduced simplicial sets

for $\infty$ -algebras general
on monoids

on simplicial T-algebras , on homotopy T-algebra s

on algebas over a monad

on algebras over an operad ,

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 operads , for Segal operads

on algebras over an operad ,

on modules over an algebra over an operad

on dendroidal sets , for dendroidal complete Segal spaces , for dendroidal Cartesian fibrations

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

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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 October 8, 2013 at 20:35:15.
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