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
on strict ∞-categories?
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
For $C$ a monoidal model category there is under mild conditions a natural model category structure on its category of monoids.
For $C$ a monoidal category with all colimits, its category of monoids comes equipped (as discussed there) with a free functor/forgetful functor adjunction
Typically one uses on $Mon(C)$ the transferred model structure along this adjunction, if it exists.
If $C$ is monoidal model category that
satisfies the monoid axiom in a monoidal 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.
This is part of (SchwedeShipley, theorem 4.1).
If the symmetric monoidal model category $C$
its unit is cofibrant
it has a suitable interval object
then the transferred model structure on monoids exists.
Regard monoids a algebras over an operad for the associative operad. Then apply the existence results discussed at model structure on algebras over an operad. See there for more details.
Suppose the transferred model structure exists on $Mon(C)$. By the discussion of free monoids at category of monoids we have that then pushouts of the form
exist in $Mon(C)$, for all $f : A \to B$ in $C$
Let the monoidal model category $C$ be
and satisfy the monoid axiom in a monoidal model category.
If $f : A\to B$ is an acyclic cofibration in the model structure on $C$, then the pushout $X \to P$ as above is a weak equivalence in $Mon(C)$.
This is SchwedeShipley, lemma 6.2.
Use the description of the pushout as a transfinite composite of pushouts as described at category of monoids in the section free and relative free monoids.
One sees that the pushout product axiom implies that all the intermediate pushouts produce acyclic cofibrations and the monoid axiom in a monoidal model category implies then that each $P_{n-1} \to P_n$ is a weak equivalence. Moreover, all these moprhisms are of the kind used in the monoid axioms, so also their transfinite composition is a weak equivalence.
Under mild conditions on $C$ the model structure on monoids in $C$ is Quillen equivalent to that of A-infinity algebras in $C$. See model structure on algebras over an operad for details.
model structure on monoids in a monoidal model category
model structure on commutative monoids in a symmetric monoidal model category
Stefan Schwede, Brooke Shipley, Algebras and modules in monoidal model categories Proc. London Math. Soc. (2000) 80(2): 491-511 (pdf)
David White, Model Structures on Commutative Monoids in General Model Categories (arXiv:1403.6759)