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.
(Schwede-Shipley 00, def. 3.3.).
In particular, the axiom in def. 1 says that for every object $X$ the functor $X \otimes (-)$ sends acyclic cofibrations to weak equivalences.
Let $C$ be a
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).
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)
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:
(Schwede-Shipley 00, section 5, MMSS 00, theorem 12.1 (iii))
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)