model category

for ∞-groupoids

# 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 $I$;

• the cofibrations are effective monomorphisms.

## 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

A standard textbook reference is section 12 of

• Hirschhorn, Model categories and their localizations

In the context of algebraic model categories related discussion is in

Revised on November 29, 2014 08:02:51 by Tim Porter (2.26.23.221)