nLab cellular model category

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

• Top with the standard Quillen model structure on topological spaces (same for pointed topological spaces)

• SSet with the standard Quillen model structure on simplicial sets (same for pointed simplicial sets).

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.

Related concepts

• the notions of cellular categories and of cellular model categories are vaguely related, but not as systematically as the (independently chosen) terminology might suggest

• combinatorial model category

References

Textbook account:

• Philip Hirschhorn, Section 12 of: Model Categories and Their Localizations, AMS Math. Survey and Monographs Vol 99 (2002) [ISBN:978-0-8218-4917-0, pdf toc, pdf]

Discussion in the context of algebraic model categories:

• Emily Riehl, Cellularity in algebraic model structures (blog post)