nLab
cellular model category

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\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 II and a set of generating acyclic cofibrations JJ, such that

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

  • the domain objects of the elements of JJ are small objects relative to II;

  • the cofibrations are effective monomorphisms.

Examples

For CC a cellular model category we have that

Applications

For cellular model categories CC that are left proper model categories all left Bousfield localizations L SCL_S C at any set SS 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 June 26, 2014 00:59:59 by Zhen Lin (131.111.24.211)