# nLab monoidal model category

model category

## Model structures

for ∞-groupoids

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

#### Monoidal categories

monoidal categories

## With traces

• trace

• traced monoidal category?

# 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 $Qe\otimes X\to e\otimes X\cong X$ is a weak equivalence, where $e$ is the unit object of the monoidal structure and $Qe\to e$ is a cofibrant 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 $ℰ$ 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 ${ℰ}^{BG}$ of objects in $ℰ$ equipped with a $G$-action, for which the forgetful functor

${ℰ}^{BG}\to ℰ$\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 ${ℰ}^{BG}$ is discussed for insztance in

Revised on June 8, 2013 22:50:45 by Adeel Khan (129.173.251.24)