on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
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.
This appears as SchwedeShipley, def. 3.3..
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).
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 $(F \dashv U) : Mon(C) \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} C$ exists on its category of monoids and hence provides a model structure on monoids.
This is part of (SchwedeShipley, theorem 4.1).
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
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 Gamma-spaces?
This is in (SchwedeShipley, section 5).