nLab
monoidal model category

Idea

A monoidal model category is a model category which is also a closed monoidal category in a compatible way. In particular, its homotopy category inherits a closed monoidal structure.

Definition

A (symmetric) monoidal model category is a category equipped with

• the structure of a closed symmetric monoidal category

• the structure of a model category;

such that

• the pushout-product axiom is satisfied, and

• For any cofibrant object $X$, the map $Qe\otimes X\to e\otimes X\cong X$ is a weak equivalence, where $e$ is the unit object of the monoidal structure and $Qe\to e$ is a cofibrant replacement for it. This is automatically satisfied if $e$ is cofibrant, as it is in most (but not all) cases.

Properties

The central fact about a monoidal model category is that its homotopy category inherits a closed monoidal structure.

Examples

• A nice category of topological spaces with cartesian product and the usual (Quillen) model structure.

• The category of simplicial sets with cartesian product and the usual (Quillen) model structure.

• The category Cat with cartesian product and the folk model structure.

• The category Gray? of strict 2-categories with the Gray tensor product and the Lack model structure?.

• The category of chain complexes with the usual tensor product of chain complexes and the projective model structure.

• The category of pointed topological spaces or simplicial sets with the smash product.

• Any of many modern model categories of spectra. The standard example of a monoidal model category whose unit is not cofibrant is the category of EKMM S-modules.

References

Mark Hovey, Model Categories