nLab
monoid axiom in a monoidal model category

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 fibrations with any object is a weak equivalence.

This appears as SchwedeShipley, def. 3.3..

Properties

Lemma

Let $C$ be a

• cofibrantly generated model category;

• symmetric monoidal category;

• monoidal model category.

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

• it is a cofibrantly generated model category;

• all objects are small objects,

then the transferred model structure along the free functor/forgetful functor adjunction $(F⊣U):\mathrm{Mon}(C)\stackrel{\stackrel{F}{\leftarrow }}{\underset{U}{\to }}C$ 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 thatt 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

• the standard model structure on simplicial sets;

• the projective model structure on chain complexes for connective chanin complexes with fibrations the surjections in positibe degree

• the model structure on symmetric spectra

• the model structure on Gamma-spaces?

This is in (SchwedeShipley, section 5).

References

• Stefan Schwede, Brooke Shipley, Algebras and modules in monoidal model categories Proc. London Math. Soc. (2000) 80(2): 491-511 (pdf)