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?
A monoidal model category is a model category which is also a closed monoidal category in a compatible way. In particular, its homotopy category inherits a closed monoidal structure.
A (symmetric) monoidal model category is a category equipped with
the structure of a closed symmetric monoidal category
the structure of a model category;
such that
the pushout-product axiom is satisfied, and
For any cofibrant object $X$, the map $Q e \otimes X \to e\otimes X \cong X$ is a weak equivalence, where $e$ is the unit object of the monoidal structure and $Q e\to e$ is a cofibrant resolution for it. This is automatically satisfied if $e$ is cofibrant, as it is in most (but not all) cases.
The central fact about a monoidal model category is that its homotopy category inherits a closed monoidal structure.
A nice category of topological spaces with cartesian product and the usual (Quillen) model structure.
The category of simplicial sets with cartesian product and the usual (Quillen) model structure.
The category Cat with cartesian product and the folk model structure.
The category Gray of strict 2-categories with the Gray tensor product and the Lack model structure?.
The category of chain complexes with the usual tensor product of chain complexes and the projective model structure.
The category of pointed compactly generated topological spaces or simplicial sets with the smash product.
Any of many modern model categories of spectra. The standard example of a monoidal model category whose unit is not cofibrant is the category of EKMM S-modules.
Let $\mathcal{E}$ be a category equipped with the structure of
such that
the model structure is cofibrantly generated;
the tensor unit $I$ is cofibrant.
Under these conditions there is for each finite group $G$ the structure of a monoidal model category on the category $\mathcal{E}^{\mathbf{B}G}$ of objects in $\mathcal{E}$ equipped with a $G$-action, for which the forgetful functor
preserves and reflects fibrations and weak equivalences.
See for instance (BergerMoerdijk 2.5).
See model structure on monoids in a monoidal model category.
A general standard reference is
The monoidal model structure on $\mathcal{E}^{\mathbf{B}G}$ is discussed for insztance in
Relation to symmetric monoidal (infinity,1)-categories is discussed in