# nLab cellular model category

Contents

### Context

#### Model category theory

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 $\infty$-algebras

specific $\infty$-algebras

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

# Contents

## Idea

A cellular model category is a particularly convenient form of a model category.

It is similar to a combinatorial model category. (For the moment, see there for more details.)

## Definition

A cellular model category is a cofibrantly generated model category such that there is a set of generating cofibrations $I$ and a set of generating acyclic cofibrations $J$, such that:

• all domain and codomain objects of elements of $I$ are compact objects relative to $I$ (in the sense of Hirschhorn);

• the domain objects of the elements of $J$ are small objects relative to $I$;

## Examples

For $C$ a cellular model category we have that

• the functor category $[D,C]$ for any small category, $D$, with its projective global model structure on functors is again a cellular model category;

• for $c \in C$ any object, the over category $C/c$ is again a cellular model category.

## Applications

For cellular model categories $C$ that are left proper model categories all left Bousfield localizations $L_S C$ at any set $S$ of morphisms are guaranteed to exist.

## References

Textbook account:

Discussion in the context of algebraic model categories:

Last revised on August 17, 2022 at 14:23:04. See the history of this page for a list of all contributions to it.