model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
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, as does the (infinity,1)-category that it presents.
A (symmetric) monoidal model category is model category $\mathcal{C}$ equipped with the structure of a closed symmetric monoidal category $(\mathcal{C}, \otimes, I)$ such that the following two compatibility conditions are satisfied
(pushout-product axiom) For every pair of cofibrations $f \colon X \to Y$ and $f' \colon X' \to Y'$, their pushout-product, hence the induced morphism out of the cofibered coproduct over ways of forming the tensor product of these objects
is itself a cofibration, which, furthermore, is acyclic (i.e. a weak equivalence) if $f$ or $f'$ is.
(Equivalently this says that the tensor product $\otimes \colon C \times C \to C$ is a left Quillen bifunctor.)
(unit axiom) For every cofibrant object $X$ and every cofibrant resolution $\emptyset \hookrightarrow Q I \stackrel{p_I}{\longrightarrow} I$ of the tensor unit $I$, the resulting morphism
is a weak equivalence.
(internal hom Quillen adjunction)
Let $X$ be a cofibrant object, hence $\varnothing \overset{\exists !}{\to} X$ a cofibration.
In this case the pushout-product axiom (Def. ) says that the tensor product functor $X \otimes (-)$ preserves cofibrations and acyclic cofibrations. Since the ambient category is assumed to be closed monoidal category, so that this functor has a right adjoint internal hom $[X,-]$, this means that it is the left Quillen functor in a Quillen adjunction
(cofibrant tensor unit implies unit axiom)
As a special case of Rem. : If the tensor unit $I$ happens to be cofibrant, then the unit axiom in def. is already implied by the pushout-product axiom.
(Because then $Q I \to I$ is a weak equivalence between cofibrant objects and such are preserved by functors that preserve acyclic cofibrations, by Ken Brown's lemma.)
One says that a monoidal model category (Def. ) satisfies the monoid axiom if every morphism that is obtained as a transfinite composition of pushouts of tensor products $X\otimes f$ of acyclic cofibrations $f$ with any object $X$ is a weak equivalence.
(Schwede-Shipley 00, def. 3.3.).
In particular, the axiom in def. says that for every object $X$ the functor $X \otimes (-)$ sends acyclic cofibrations to weak equivalences.
For $(\mathcal{C},\otimes)$ a monoidal model category, def. , then the derived functor $\otimes^L$ of the tensor product makes the homotopy category of the model category itself into a monoidal category, such that the localization functor
is a lax monoidal functor.
Let $V$ be a monoidal model category, and consider it as a derivable category in the sense of (Shulman 11, section 8) with $V_Q$ the subcategory of cofibrant objects and $V_R=V$. Then $\otimes :V\times V\to V$ is left derivable, i.e. it preserves the $Q$-subcategories and weak equivalences. Since deriving is pseudofunctorial (and product-preserving) on the 2-category of derivable categories and left derivable functors, it follows immediately that $Ho(V)$ is monoidal; this is (Shulman 11, example 8.13).
Now let $V_0$ denote the category $V$ with its trivial derivable structure: only isomorphisms are weak equivalences, and all objects are both $Q$ and $R$. Then of course $Ho(V_0) = V$, and $V_0$ is also a pseudomonoid in derivable categories. The identity functor $Id : V_0 \to V$ is not left derivable, since it does not preserve $Q$-objects; but it is right derivable, since we took all objects in $V$ to be $R$-objects (ignoring the fibrant objects in the model structure on $V$). Of course $Id$ is strong monoidal, and this monoidality constraint can be expressed as a square in the double category of (Shulman 11) whose vertical arrows are left derivable functors and whose horizontal arrows are right derivable functors; moreover the axioms on a monoidal functor may be expressed using products and double-categorical pasting in this double category. Therefore, it is all preserved by the double pseudofunctor $Ho$; but $Ho(Id) = \gamma : V \to Ho(V)$. The only thing that is not visible to the double category is the invertibility of the monoidal constraint, and hence this is not preserved by the double pseudofunctor; thus $\gamma$ is only lax monoidal.
Let $(\mathcal{C}, \otimes, 1)$ be a monoidal model category with cofibrant tensor unit. Then the left derived functor $\otimes^L$ of the tensor product exists
and makes the homotopy category into a monoidal category $(Ho(\mathcal{C}), \otimes^L, \gamma(1))$.
Moreover, the localization functor
on the category of cofibrant objects is a strong monoidal functor with structure morphism the inverse of the above natural isomorpmism
For the left derived functor (def.) of the tensor product
to exist, it is sufficient that its restriction to the subcategory
of cofibrant objects preserves acyclic cofibrations (Ken Brown's lemma, here).
Every morphism $(f,g)$ in the product category $\mathcal{C}_{c}\times \mathcal{C}_{c}$ may be written as a composite of a pairing with an identity morphisms
Now since the pushout product (with respect to tensor product) with the initial morphism $(\ast \to c_1)$ is equivalently the tensor product
and
the pushout-product axiom (def. ) implies that on the subcategory of cofibrant objects the functor $\otimes$ preserves acyclic cofibrations. (This is why one speaks of a Quillen bifunctor, see also Hovey 99, prop. 4.3.1).
Hence $\otimes^L$ exists.
By the same decomposition and using the universal property of the localization of a category (def.) one finds that for $\mathcal{C}$ and $\mathcal{D}$ any two categories with weak equivalences (def.) then the localization of their product category is the product category of their localizations:
With this, the universal property as a localization (def.) of the homotopy category of a model category (thm.) induces associators: Let
be the natural isomorphism exhibiting the derived functors of the two possible tensor products of three objects, as shown at the top. By pasting the second with the associator natural isomorphism of $\mathcal{C}$ we obtain another such factorization for the first, as shown on the left below,
and hence by the universal property of the factorization through the derived functor, there exists a unique natural isomorphism $\alpha^L$ such as to make this composite of natural isomorphisms equal to the one shown on the right. Hence the pentagon identity satisfied by $\alpha$ implies a pentagon identity for $\alpha^L$, and so $\alpha^L$ is an associator for $\otimes^L$.
The above equation on pasting composites of natural isomorphism is equivalently just the coherence law for a monoidal functor:
A nice category of topological spaces with cartesian product and the classical model structure on topological spaces.
The category of simplicial sets with cartesian product and the classical model structure on simplicial sets.
The classical model structure on pointed topological spaces or pointed simplicial sets with the smash product of pointed objects.
If the underlying site has finite product, then both the injective and the projective, the global and the local model structure on simplicial presheaves becomes a cartesian? monoidal model category with respect to the standard closed monoidal structure on presheaves.
See at model structure on simplicial presheaves the section Closed monoidal structure.
With respect to a symmetric monoidal smash product of spectra:
(Schwede-Shipley 00, MMSS 00, theorem 12.1 (iii) with prop. 12.3)
The standard example of a monoidal model category whose unit is not cofibrant is the category of EKMM S-modules.
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?.
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 Borel model structure $\mathcal{E}^{\mathbf{B}G}$ 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.
Textbook accounts:
Mark Hovey, Def. 4.2.6 in: Model Categories, Mathematical Surveys and Monographs, Volume 63, AMS (1999) (ISBN:978-0-8218-4361-1, doi:10.1090/surv/063, pdf, Google books)
Jacob Lurie, Def. A.3.1.2 in: Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press 2009 (pup:8957, pdf)
Some relevant homotopy category background is in
The monoidal structures for a symmetric monoidal smash product of spectra are due to
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)
Further variation of the axiomatics is discussed in
The monoidal model structure on the Borel model structure $\mathcal{E}^{\mathbf{B}G}$ is discussed for instance in
Relation to symmetric monoidal (infinity,1)-categories (in particular, that every locally presentable symmetric monoidal $(\infty,1)$-category arises from a symmetric monoidal model category) is discussed in
Monoidal Reedy model structures are discussed in