# nLab monoid axiom in a monoidal model category

Contents

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.

###### Remark

In particular, the axiom in def. says that for every object $X$ the functor $X \otimes (-)$ sends acyclic cofibrations to weak equivalences.

## Properties

###### Lemma

Let $C$ be a

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

###### Theorem

If a monoidal model category satisfies the monoid axiom and

then the transferred model structure along the free-forgetful 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.

## 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

with respect to Cartesian product

with respect to tensor product of chain complexes:

and with respect to a symmetric monoidal smash product of spectra: