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\begin{document}

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\section*{monoidal model category}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{model_category_theory}{}\paragraph*{{Model category theory}}\label{model_category_theory}

[[!include model category theory - contents]]

\hypertarget{monoidal_categories}{}\paragraph*{{Monoidal categories}}\label{monoidal_categories}

[[!include monoidal categories - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{MonoidalHomotopyCategory}{Monoidal homotopy category}\dotfill \pageref*{MonoidalHomotopyCategory} \linebreak
\noindent\hyperlink{abstractly}{Abstractly}\dotfill \pageref*{abstractly} \linebreak
\noindent\hyperlink{explicitly}{Explicitly}\dotfill \pageref*{explicitly} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{classical_homotopy_theory}{Classical homotopy theory}\dotfill \pageref*{classical_homotopy_theory} \linebreak
\noindent\hyperlink{SimplicialPresheaves}{Simplicial presheaves}\dotfill \pageref*{SimplicialPresheaves} \linebreak
\noindent\hyperlink{homological_algebra_and_stable_homotopy_theory}{Homological algebra and stable homotopy theory}\dotfill \pageref*{homological_algebra_and_stable_homotopy_theory} \linebreak
\noindent\hyperlink{categorical_model_structures}{Categorical model structures}\dotfill \pageref*{categorical_model_structures} \linebreak
\noindent\hyperlink{model_structure_on_objects}{Model structure on $G$-objects}\dotfill \pageref*{model_structure_on_objects} \linebreak
\noindent\hyperlink{model_structure_on_monoids}{Model structure on monoids}\dotfill \pageref*{model_structure_on_monoids} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{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.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{MonoidalModelCategory}\hypertarget{MonoidalModelCategory}{}
A (symmetric) \textbf{monoidal model category} is [[model category]] $\mathcal{C}$ equipped with the structure of a [[closed monoidal category|closed]] [[symmetric monoidal category]] $(\mathcal{C}, \otimes, I)$ such that the following two compatibility conditions are satisfied

\begin{enumerate}%
\item ([[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

\begin{displaymath}
(X \otimes Y') \coprod_{X \otimes X'} (Y \otimes X')
\longrightarrow
Y \otimes Y'
\,,
\end{displaymath}
is itself a cofibration, which, furthermore, is acyclic if $f$ or $f'$ is.

(Equivalently this says that the [[tensor product]] $\otimes : C \times C \to C$ is a left [[Quillen bifunctor]].)


\item (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

\begin{displaymath}
Q I \otimes X \stackrel{p_I \otimes X}{\longrightarrow} I\otimes X \stackrel{\simeq}{\longrightarrow} X
\end{displaymath}


\end{enumerate}
is a weak equivalence.

\end{defn}
\begin{remark}
\label{CaseOfCofibrantTensorUnit}\hypertarget{CaseOfCofibrantTensorUnit}{}
The [[pushout-product axiom]] in def. \ref{MonoidalModelCategory} implies that for $X$ a cofibrant object, then the functor $X \otimes (-)$ is preserves cofibrations and acyclic cofibrations.

In particular if the [[tensor unit]] $I$ happens to be cofibrant, then the unit axiom in def. \ref{MonoidalModelCategory} is implied by the \hyperlink{PushoutProductAxiom}{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]]. )

\end{remark}
\begin{defn}
\label{MonoidAxiom}\hypertarget{MonoidAxiom}{}
We say a [[monoidal model category]], def. \ref{MonoidalModelCategory}, satisfies the \textbf{[[monoid axiom]]}, def. \ref{MonoidalModelCategory}, 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.

\end{defn}
(\hyperlink{SchwedeShipley00}{Schwede-Shipley 00, def. 3.3.}).

\begin{remark}
\label{}\hypertarget{}{}
In particular, the axiom in def. \ref{MonoidAxiom} says that for every object $X$ the functor $X \otimes (-)$ sends acyclic cofibrations to weak equivalences.

\end{remark}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{MonoidalHomotopyCategory}{}\subsubsection*{{Monoidal homotopy category}}\label{MonoidalHomotopyCategory}

\hypertarget{abstractly}{}\paragraph*{{Abstractly}}\label{abstractly}

\begin{prop}
\label{LocalizationFunctorIsLaxMonoidal}\hypertarget{LocalizationFunctorIsLaxMonoidal}{}
For $(\mathcal{C},\otimes)$ a monoidal model category, def. \ref{MonoidalModelCategory}, then the [[derived functor]] $\otimes^L$ of the tensor product makes the [[homotopy category of a model category|homotopy category of the model category]] itself into a [[monoidal category]], such that the [[localization]] functor

\begin{displaymath}
\gamma
   \;\colon\;
  (\mathcal{C},\otimes)
    \longrightarrow
  (Ho(\mathcal{C}), \otimes^L)
\end{displaymath}
is a [[lax monoidal functor]].

\end{prop}
\begin{proof}
Let $V$ be a monoidal model category, and consider it as a \emph{[[derivable category]]} in the sense of (\hyperlink{Shulman11}{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 [[pseudofunctor|pseudofunctorial]] (and product-preserving) on the [[2-category]] of [[derivable categories]] and [[left derived functor|left derivable functors]], it follows immediately that $Ho(V)$ is monoidal; this is (\hyperlink{Shulman11}{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 \emph{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 functor|strong monoidal]], and this monoidality constraint can be expressed as a square in the [[double category]] of (\hyperlink{Shulman11}{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 functor|lax monoidal]].

\end{proof}
\hypertarget{explicitly}{}\paragraph*{{Explicitly}}\label{explicitly}

\begin{prop}
\label{MonoidalStructureOnHomotopyCategoryOfMonoidalModelCategory}\hypertarget{MonoidalStructureOnHomotopyCategoryOfMonoidalModelCategory}{}
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

\begin{displaymath}
\itexarray{
    \mathcal{C}_c\times \mathcal{C}_c
      &\overset{\otimes}{\longrightarrow}&
    \mathcal{C}_c
    \\
    {}^{\mathllap{\gamma \times \gamma}}\downarrow 
      &\swArrow_{\mu^{-1}}& 
    \downarrow^{\mathrlap{\gamma}}
    \\
    Ho(\mathcal{C})\times Ho(\mathcal{C})
     &\overset{\otimes^L}{\longrightarrow}&
    Ho(\mathcal{C})
  }
\end{displaymath}
and makes the [[homotopy category of a model category|homotopy category]] into a [[monoidal category]] $(Ho(\mathcal{C}), \otimes^L, \gamma(1))$.

Moreover, the [[localization]] functor

\begin{displaymath}
\gamma
  \;\colon\;
  (\mathcal{C}_c, \otimes, 1)
   \longrightarrow
  (Ho(\mathcal{C}), \otimes^L , \gamma(1))
\end{displaymath}
on the [[category of cofibrant objects]] is a [[strong monoidal functor]] with structure morphism the inverse of the above natural isomorpmism

\begin{displaymath}
\mu_{X,Y}
  \;\colon\;
  \gamma(X)\otimes^L \gamma(Y)
    \longrightarrow
  \gamma(X \otimes Y)
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
For the [[left derived functor]] (\href{Introduction+to+Stable+homotopy+theory+--+P#LeftAndRightDerivedFunctorsOnModelCategories}{def.}) of the tensor product

\begin{displaymath}
\otimes
  \;
  \mathcal{C}\times \mathcal{C}
    \longrightarrow
  \mathcal{C}
\end{displaymath}
to exist, it is sufficient that its restriction to the subcategory

\begin{displaymath}
(\mathcal{C} \times \mathcal{C})_c
  \simeq
  \mathcal{C}_c \times \mathcal{C}_c
\end{displaymath}
of cofibrant objects preserves acyclic cofibrations ([[Ken Brown's lemma]], \href{Introduction+to+Stable+homotopy+theory+--+P#KenBrownLemma}{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

\begin{displaymath}
(f,g)
    \;\colon\;
  (c_1, d_1)
    \overset{(id_{c_1},g)}{\longrightarrow}
  (c_1,d_2)
    \overset{(f,id_{c_2})}{\longrightarrow}
  (c_2,d_2)
  \,.
\end{displaymath}
Now since the [[pushout product]] (with respect to tensor product) with the initial morphism $(\ast \to c_1)$ is equivalently the tensor product

\begin{displaymath}
(\ast \to c_1) \Box_{\otimes} g
    \;\simeq\;
  id_{c_1} \otimes g
\end{displaymath}
and

\begin{displaymath}
f \Box_{\otimes} (\ast \to c_2)
    \;\simeq\;
  f \otimes id_{c_2}
\end{displaymath}
the [[pushout-product axiom]] (def. \ref{MonoidalModelCategory}) implies that on the subcategory of cofibrant objects the functor $\otimes$ preserves acyclic cofibrations. (This is why one speaks of a \emph{[[Quillen bifunctor]]}, see also \hyperlink{Hovey99}{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 (\href{Introduction+to+Stable+homotopy+theory+--+P#HomotopyCategoryOfACategoryWithWeakEquivalences}{def.}) one finds that for $\mathcal{C}$ and $\mathcal{D}$ any two [[categories with weak equivalences]] (\href{Introduction+to+Stable+homotopy+theory+--+P#CategoryWithWeakEquivalences}{def.}) then the [[localization]] of their [[product category]] is the product category of their localizations:

\begin{displaymath}
(\mathcal{C} \times \mathcal{D})[(W_{\mathcal{C}} \sqcup W_{\mathcal{D}})^{-1}]
  \simeq
  (\mathcal{C}[W^{-1}_{\mathcal{C}}])
  \times
  (\mathcal{D}[W^{-1}_{\mathcal{D}}])
  \,.
\end{displaymath}
With this, the [[universal property]] as a [[localization]] (\href{Introduction+to+Stable+homotopy+theory+--+P#HomotopyCategoryOfACategoryWithWeakEquivalences}{def.}) of the [[homotopy category of a model category]] (\href{Introduction+to+Stable+homotopy+theory+--+P#UniversalPropertyOfHomotopyCategoryOfAModelCategory}{thm.}) induces [[associators]]: Let

\begin{displaymath}
\itexarray{
    \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c
      &\overset{((-)\otimes(-))\otimes (-)}{\longrightarrow}&
    \mathcal{C}
    \\
    {}^{\mathllap{\gamma_{\mathcal{C} \times \mathcal{C} \times \mathcal{C}}}}\downarrow
      &\swArrow_{\eta}&
    \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}}
    \\
    Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C})
      &\overset{((-)\otimes^L(-))\otimes^L (-)}{\longrightarrow}&
    Ho(\mathcal{C})
  }
  \;\;\;\,,\;\;\;\;\;
  \itexarray{
    \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c
      &\overset{ (-) \otimes ( (-) \otimes (-) ) }{\longrightarrow}&
    \mathcal{C}
    \\
    {}^{\mathllap{\gamma_{\mathcal{C} \times \mathcal{C} \times \mathcal{C}}}}\downarrow
      &\swArrow_{\eta'}&
    \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}}
    \\
    Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C})
      &\overset{ (-) \otimes^L ( (-) \otimes^L (-) )  }{\longrightarrow}&
    Ho(\mathcal{C})
  }
\end{displaymath}
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,

\begin{displaymath}
(\star)
  \;\;\;\;\;\;
  \itexarray{
    \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c
      &\overset{((-)\otimes(-))\otimes (-)}{\longrightarrow}&
    \mathcal{C}
    \\
    {}^{\mathllap{=}}\downarrow 
      &\swArrow_{\alpha}&
    \downarrow^{\mathrlap{=}}
    \\
    \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c
      &\overset{ (-) \otimes ( (-) \otimes (-) ) }{\longrightarrow}&
    \mathcal{C}
    \\
    {}^{\mathllap{\gamma_{\mathcal{C} \times \mathcal{C} \times \mathcal{C}}}}\downarrow
      &\swArrow_\eta&
    \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}}
    \\
    Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C})
      &\overset{ (-) \otimes^L ( (-) \otimes^L (-) )  }{\longrightarrow}&
    Ho(\mathcal{C})
  }
  \;\;\;\;\;\;
  \simeq
  \;\;\;\;\;\;
  \itexarray{
    \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c
      &\overset{((-)\otimes(-))\otimes (-)}{\longrightarrow}&
    \mathcal{C}
    \\
    {}^{\mathllap{\gamma_{\mathcal{C} \times \mathcal{C} \times \mathcal{C}}}}\downarrow
      &\swArrow_{\eta'}&
    \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}}
    \\
    Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C})
      &\overset{((-)\otimes^L(-))\otimes^L (-)}{\longrightarrow}&
    Ho(\mathcal{C})
    \\
    {}^{\mathllap{=}}\downarrow 
      &\swArrow_{\alpha^L}&
    \downarrow^{\mathrlap{=}}
    \\
    Ho(\mathcal{C})\times Ho(\mathcal{C})\times Ho(\mathcal{C})
      &\underset{(-)\otimes^L((-)\otimes^L (-))}{\longrightarrow}&
    Ho(\mathcal{C})
  }
\end{displaymath}
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:

\begin{displaymath}
\itexarray{
    (\gamma(X) \otimes^L \gamma(Y)) \otimes^L \gamma(Z)
     &\overset{\alpha^L_{\gamma(X), \gamma(Y), \gamma(Z)}}{\longrightarrow}&
    \gamma(X) \otimes^L (\gamma(Y) \otimes^L \gamma(Z))
    \\
    {}^{\mathllap{\eta'}}\uparrow
      &&
    \uparrow^{\mathrlap{\eta}}
    \\
    \gamma( (X \otimes Y) \otimes Z )
     &\underset{\gamma(\alpha)}{\longrightarrow}&
   \gamma(X \otimes (Y \otimes Z))
  }
  \,.
\end{displaymath}
\end{proof}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{classical_homotopy_theory}{}\subsubsection*{{Classical homotopy theory}}\label{classical_homotopy_theory}

\begin{itemize}%
\item A [[nice category of spaces|nice category of topological spaces]] with cartesian product and the [[classical model structure on topological spaces]].


\item The category of [[simplicial sets]] with cartesian product and the [[classical model structure on simplicial sets]].


\item The [[classical model structure on pointed topological spaces]] or pointed simplicial sets with the [[smash product]] of pointed objects.



\end{itemize}
\hypertarget{SimplicialPresheaves}{}\subsubsection*{{Simplicial presheaves}}\label{SimplicialPresheaves}

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 \emph{[[model structure on simplicial presheaves]]} the section \emph{\href{https://ncatlab.org/nlab/show/model+structure+on+simplicial+presheaves#MonoidalStructure}{Closed monoidal structure}}.

\hypertarget{homological_algebra_and_stable_homotopy_theory}{}\subsubsection*{{Homological algebra and stable homotopy theory}}\label{homological_algebra_and_stable_homotopy_theory}

\begin{itemize}%
\item The category of [[chain complex]]es with the usual [[tensor product]] of chain complexes and the [[model structure on chain complexes|projective model structure]].

\end{itemize}
With respect to a [[symmetric monoidal smash product of spectra]]:

\begin{itemize}%
\item the [[model structure on symmetric spectra]]


\item the [[model structure on orthogonal spectra]]


\item the [[model structure for excisive functors]]



\end{itemize}
(\hyperlink{SchwedeShipley00}{Schwede-Shipley 00}, \hyperlink{MMSS00}{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.

\hypertarget{categorical_model_structures}{}\subsubsection*{{Categorical model structures}}\label{categorical_model_structures}

\begin{itemize}%
\item The category [[Cat]] with cartesian product and the [[folk model structure]].


\item The category [[Gray]] of [[strict 2-category|strict 2-categories]] with the [[Gray tensor product]] and the [[model structure on 2-categories|Lack model structure]].



\end{itemize}
\hypertarget{model_structure_on_objects}{}\subsubsection*{{Model structure on $G$-objects}}\label{model_structure_on_objects}

\begin{udef}
Let $\mathcal{E}$ be a [[category]] equipped with the structure of

\begin{itemize}%
\item a [[closed monoidal category|closed]] [[symmetric monoidal category]];


\item a [[monoidal model category]];



\end{itemize}
such that

\begin{itemize}%
\item the model structure is [[cofibrantly generated model category|cofibrantly generated]];


\item the tensor unit $I$ is cofibrant.



\end{itemize}
\end{udef}
\begin{uprop}
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]]

\begin{displaymath}
\mathcal{E}^{\mathbf{B}G} \to \mathcal{E}
\end{displaymath}
preserves and reflects fibrations and weak equivalences.

\end{uprop}
See for instance (\hyperlink{BergerMoerdijkResolution}{BergerMoerdijk 2.5}).

\hypertarget{model_structure_on_monoids}{}\subsubsection*{{Model structure on monoids}}\label{model_structure_on_monoids}

See [[model structure on monoids in a monoidal model category]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[monoidal localization]]


\item [[monoidal Quillen adjunction]]


\item [[model structure on monoids in a monoidal model category]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

A textbook reference is

\begin{itemize}%
\item [[Mark Hovey]], chapter 4 of \emph{Model Categories} Mathematical Surveys and Monographs, Volume 63, AMS (1999) (\href{https://www.math.rochester.edu/people/faculty/doug/otherpapers/hovey-model-cats.pdf}{pdf}, \href{http://books.google.co.uk/books?id=Kfs4uuiTXN0C&printsec=frontcover}{Google books})

\end{itemize}
Some relevant homotopy category background is in

\begin{itemize}%
\item [[Mike Shulman]], \emph{Comparing composites of left and right derived functors}, New York Journal of Mathematics Volume 17 (2011) 75-125 (\href{http://arxiv.org/abs/0706.2868}{arXiv:0706.2868}, \href{http://nyjm.albany.edu/j/2011/17-5.html}{publisher})

\end{itemize}
The monoidal structures for a [[symmetric monoidal smash product of spectra]] are due to

\begin{itemize}%
\item [[Stefan Schwede]], [[Brooke Shipley]], \emph{Algebras and modules in monoidal model categories} Proc. London Math. Soc. (2000) 80(2): 491-511 (\href{http://www.math.uic.edu/~bshipley/monoidal.pdf}{pdf})


\item [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]], part III of \emph{[[Model categories of diagram spectra]]}, Proceedings London Mathematical Society Volume 82, Issue 2, 2000 (\href{http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf}{pdf}, \href{http://plms.oxfordjournals.org/content/82/2/441.short?rss=1&ssource=mfc}{publisher})



\end{itemize}
Further variation of the axiomatics is discussed in

\begin{itemize}%
\item [[Michael Batanin]], [[Clemens Berger]], \emph{Homotopy theory for algebras over polynomial monads} (\href{https://arxiv.org/abs/1305.0086}{arXiv:1305.0086})

\end{itemize}
The monoidal model structure on $\mathcal{E}^{\mathbf{B}G}$ is discussed for instance in

\begin{itemize}%
\item [[Clemens Berger]], [[Ieke Moerdijk]], \emph{The Boardman-Vogt resolution of operads in monoidal model categories}, Topology 45 (2006), 807--849.

\end{itemize}
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

\begin{itemize}%
\item [[Thomas Nikolaus]], [[Steffen Sagave]], \emph{Presentably symmetric monoidal infinity-categories are represented by symmetric monoidal model categories} (\href{http://arxiv.org/abs/1506.01475}{arXiv:1506.01475})

\end{itemize}
[[!redirects monoidal model categories]] [[!redirects monoidal model structure]] [[!redirects monoidal model structures]] [[!redirects symmetric monoidal model structure]] [[!redirects symmetric monoidal model structures]] [[!redirects symmetric monoidal model category]] [[!redirects symmetric monoidal model categories]]



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