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 cofibrations with any object is a weak equivalence.

(Schwede-Shipley 00, def. 3.3.).

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

• cofibrantly generated model category;

• symmetric monoidal category;

• monoidal model category.

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

(Schwede-Shipley 00, 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-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.

(Schwede-Shipley 00, 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

with respect to Cartesian product

• the standard model structure on simplicial sets;

with respect to tensor product of chain complexes:

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

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

• the model structure on symmetric spectra

• the model structure on orthogonal spectra

• the model structure for excisive functors

• the model structure on Gamma-spaces

(Schwede-Shipley 00, section 5, MMSS 00, theorem 12.1 (iii))

References

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

• Michael Mandell, Peter May, Stefan Schwede, Brooke Shipley, part III of Model categories of diagram spectra, Proceedings London Mathematical Society Volume 82, Issue 2, 2000 (pdf, publisher)