# nLab monoidal model category

model category

## Model structures

for ∞-groupoids

### for $(\infty,1)$-sheaves / $\infty$-stacks

#### Monoidal categories

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 $X$, the map $Q e \otimes X \to e\otimes X \cong X$ is a weak equivalence, where $e$ is the unit object of the monoidal structure and $Q e\to e$ is a cofibrant resolution 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

### Model structure on $G$-objects

###### Assumption

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

such that

• the model structure is cofibrantly generated;

• the tensor unit $I$ is cofibrant.

###### Proposition

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

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

preserves and reflects fibrations and weak equivalences.

See for instance (BergerMoerdijk 2.5).

## References

A general standard reference is

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

Relation to symmetric monoidal (infinity,1)-categories is discussed in

Revised on July 18, 2015 09:36:12 by Urs Schreiber (94.118.156.166)