related by the Dold-Kan correspondence
category with duals (list of them)
dualizable object (what they have)
symmetric monoidal (∞,1)-category of spectra
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.
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.
In particular, the axiom in def. 1 says that for every object the functor sends acyclic cofibrations to weak equivalences.
Let be a
Then if the monoid axiom holds for the set of generating acyclic cofibrations it holds for all acyclic cofibrations.
If a monoidal model category satisfies the monoid axiom and
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: