nLab
monoidal 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

Monoidal categories

Contents

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

such that

  • the pushout-product axiom is satisfied, and

  • For any cofibrant object XX, the map QeXeXXQ e \otimes X \to e\otimes X \cong X is a weak equivalence, where ee is the unit object of the monoidal structure and QeeQ e\to e is a cofibrant resolution for it. This is automatically satisfied if ee 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

Model structure on GG-objects

Assumption

Let \mathcal{E} be a category equipped with the structure of

such that

Proposition

Under these conditions there is for each finite group GG the structure of a monoidal model category on the category BG\mathcal{E}^{\mathbf{B}G} of objects in \mathcal{E} equipped with a GG-action, for which the forgetful functor

BG \mathcal{E}^{\mathbf{B}G} \to \mathcal{E}

preserves and reflects fibrations and weak equivalences.

See for instance (BergerMoerdijk 2.5).

Model structure on monoids

See model structure on monoids in a monoidal model category.

References

A general standard reference is

The monoidal model structure on BG\mathcal{E}^{\mathbf{B}G} is discussed for insztance in

Revised on October 12, 2013 06:38:56 by Adeel Khan (132.252.62.187)