# nLab monoid axiom in a monoidal model category

model category

## Model structures

for ∞-groupoids

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

#### Monoidal categories

monoidal categories

## In higher category theory

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

The monoid axiom is an extra condition on a monoidal model category that helps to make its model structure on monoids in a monoidal model category exist and be well behaved.

## Definition

###### Definition

We say a monoidal model category satisfies the monoid axiom if every morphism that is obtained as a transfinite composition of pushouts of tensor products of acyclic cofibrations with any object is a weak equivalence.

This appears as SchwedeShipley, def. 3.3..

## Properties

###### Lemma

Let $C$ be a

Then if the monoid axiom hold for the set of generating acyclic cofibrations it holds for all acyclic cofibrations.

This is (SchwedeShipley, lemma 3.5).

###### Theorem

If a monoidal model category satisfies the monoid axiom and

then the transferred model structure along the free functor/forgetful functor adjunction $(F \dashv U) : Mon(C) \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} C$ exists on its category of monoids and hence provides a model structure on monoids.

This is part of (SchwedeShipley, theorem 4.1).

## Examples

###### Proposition

Monoidal model categories that satisfy the monoid axiom (as well as the other conditions sufficient for the above theorem on the existence of transferred model structures on categories of monoids) include

This is in (SchwedeShipley, section 5).

## References

Revised on October 19, 2014 23:21:08 by Anonymous Coward (82.82.235.75)