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\begin{document}

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\section*{model structure on dendroidal sets}

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

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

[[!include model category theory - contents]]

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{overview}{Overview}\dotfill \pageref*{overview} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{special_morphisms}{Special morphisms}\dotfill \pageref*{special_morphisms} \linebreak
\noindent\hyperlink{the_model_structure}{The model structure}\dotfill \pageref*{the_model_structure} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{establishing_the_model_structure}{Establishing the model structure}\dotfill \pageref*{establishing_the_model_structure} \linebreak
\noindent\hyperlink{StatementOfTheModelStructure}{Statement}\dotfill \pageref*{StatementOfTheModelStructure} \linebreak
\noindent\hyperlink{ProofOfTheModelStructure}{Proof}\dotfill \pageref*{ProofOfTheModelStructure} \linebreak
\noindent\hyperlink{CharacterizationOfTheFibrantObjects}{Characterization of the fibrations}\dotfill \pageref*{CharacterizationOfTheFibrantObjects} \linebreak
\noindent\hyperlink{MonoidalModelCategoryStructure}{Monoidal model category structure}\dotfill \pageref*{MonoidalModelCategoryStructure} \linebreak
\noindent\hyperlink{Enrichment}{Other enrichments of the underlying category}\dotfill \pageref*{Enrichment} \linebreak
\noindent\hyperlink{relation_to_other_model_structures}{Relation to other model structures}\dotfill \pageref*{relation_to_other_model_structures} \linebreak
\noindent\hyperlink{model_structure_for_quasicategories}{Model structure for quasi-categories}\dotfill \pageref*{model_structure_for_quasicategories} \linebreak
\noindent\hyperlink{RelationToModelStructureForDendroidalCompleteSegal}{Model structure for dendroidal complete Segal spaces}\dotfill \pageref*{RelationToModelStructureForDendroidalCompleteSegal} \linebreak
\noindent\hyperlink{RelationToModelStrictureOnSimplicialOperads}{Model structure on simplicial operads}\dotfill \pageref*{RelationToModelStrictureOnSimplicialOperads} \linebreak
\noindent\hyperlink{model_structure_on_operads}{Model structure on $Set$-Operads}\dotfill \pageref*{model_structure_on_operads} \linebreak
\noindent\hyperlink{model_structure_for_symmetric_monoidal_categories}{Model structure for symmetric monoidal $(\infty,1)$-categories}\dotfill \pageref*{model_structure_for_symmetric_monoidal_categories} \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}

The [[Denis-Charles Cisinski|Cisinski]]--[[Ieke Moerdijk|Moerdijk]] [[model category]] structure on the [[category]] [[dSet]] of [[dendroidal sets]] models [[(∞,1)-operads]] in generalization of the way the [[Andre Joyal|Joyal]] [[model structure on simplicial sets]] models [[(∞,1)-category|(∞,1)-categories]].

\hypertarget{overview}{}\subsection*{{Overview}}\label{overview}

We have the following diagram of [[model category|model categories]]:

\begin{displaymath}
\itexarray{
    SSet\text{-}Operad 
     &\stackrel{\simeq}{\to}& 
    dSet
     &\stackrel{\simeq}{\to}& 
    dSpaces
    &&&&& models\;for\;(\infty,1)\text{-}operads
    \\
    \uparrow && \uparrow && \uparrow
    \\
    SSet\text{-}Cat 
     &\stackrel{\simeq}{\to}& 
    sSet
     &\stackrel{\simeq}{\to}& 
    sSpaces   
    &&&&&
    models\;for\;(\infty,1)\text{-}categories
  }
  \,,
\end{displaymath}
where the entries are

\begin{itemize}%
\item the category $SSet Cat$ of [[simplicially enriched category|simplicially enriched categories]] equipped with the [[Julie Bergner|Bergner]] model structure;


\item the category [[SSet]] of [[simplicial set]]s equipped with the [[model structure on simplicial sets|Joyal model structure]] for [[quasi-category|quasi-categories]];


\item the category $dSet$ of [[dendroidal set]]s



\end{itemize}
and where

\begin{itemize}%
\item the horizontal morphisms are [[Quillen equivalence]]s


\item the vertical morphisms are [[reflective sub-(infinity,1)-category|homotopy full embeddings]].



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

\hypertarget{special_morphisms}{}\subsubsection*{{Special morphisms}}\label{special_morphisms}

Recall from the entry on [[dendroidal set]]s the definition of inner and outer faces, boundaries and inner and outer horns.

The following definition are the obvious generalizations of the corresponding notions for the [[model structure on simplicial sets]], in particular for the [[model structure for quasi-categories]].

\begin{defn}
\label{InnerAnodyneExtension}\hypertarget{InnerAnodyneExtension}{}
The class of morphisms in $dSet$ generated from the inner horn inclusions $\Lambda^e \Omega[T] \to \Omega[T]$ under

\begin{itemize}%
\item [[pushout]]


\item [[transfinite composition]]


\item [[retract]]s



\end{itemize}
is called the \textbf{inner [[anodyne extensions]]}.

\end{defn}
\begin{defn}
\label{NormalMonomorphism}\hypertarget{NormalMonomorphism}{}
The class of morphisms in $dSet$ generated from boundary inclusions $\delta \Omega[T] \to \Omega[T]$ under

\begin{itemize}%
\item [[pushout]]


\item [[transfinite composition]]


\item [[retracts]]



\end{itemize}
is called the \textbf{\href{http://ncatlab.org/nlab/show/dendroidal%20set#NormalMonomorphisms}{normal monomorphisms}}.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
A [[morphism]] $A \to B$ in $dSet$ is an \textbf{inner Kan fibration} if it has the [[right lifting property]] with respect to all inner horn inclusions.

\begin{displaymath}
\itexarray{
    \Lambda^e[T] &\to& A
     \\
    \downarrow && \downarrow
    \\
    \Omega[T] &\to& B
  }
\end{displaymath}
or equivalently with respect to the class of inner anodyne extensions.

\end{defn}
\begin{defn}
\label{QuasiOperad}\hypertarget{QuasiOperad}{}
A dendroidal set $X$ is an \textbf{inner Kan complex} or \textbf{quasi-operad} if the canonical morphism $X\to {*}$ to the [[terminal object]] is an inner Kan fibration.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
A morphism $A \to B$ of dendroidal sets is an \textbf{acyclic fibration} if it has the [[right lifting property]] with respect to all normal monomorphisms.

\end{defn}
\begin{defn}
\label{Isofibration}\hypertarget{Isofibration}{}
A morphism $f : X \to Y$ of dendroidal sets is called an \textbf{isofibration} if it

\begin{enumerate}%
\item it is an inner Kan fibration;


\item the morphism of operads $\tau_d(f) : \tau_d(X) \to \tau_d(Y)$ is a fibration in the [[canonical model structure on operads]], hence the underlying [[functor]] of categories is an [[isofibration]].



\end{enumerate}
\end{defn}
\hypertarget{the_model_structure}{}\subsubsection*{{The model structure}}\label{the_model_structure}

\begin{defn}
\label{ModelStructureOnDendroidalSets}\hypertarget{ModelStructureOnDendroidalSets}{}
On the category of [[dendroidal sets]] let

\begin{itemize}%
\item the cofibrations be the \hyperlink{NormalMonomorphism}{normal monomorphisms};


\item the fibrant objects are the weak Kan complexes/\hyperlink{QuasiOperad}{quasi-operads};


\item the fibrations between fibrant objects are precisely the \hyperlink{Isofibration}{isofibrations}.


\item the weak equivalences form the smallest class of maps that satisfy

\begin{itemize}%
\item [[category with weak equivalences|2-out-of-3]]


\item every \hyperlink{InnerAnodyneExtension}{inner anodyne extension} is a weak equivalence


\item every acyclic fibration between quasi-operads is a weak equivalence.



\end{itemize}


\end{itemize}
\end{defn}
\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\hypertarget{establishing_the_model_structure}{}\subsubsection*{{Establishing the model structure}}\label{establishing_the_model_structure}

\hypertarget{StatementOfTheModelStructure}{}\paragraph*{{Statement}}\label{StatementOfTheModelStructure}

\begin{theorem}
\label{TheModelStructureExistenceAndBasicProperties}\hypertarget{TheModelStructureExistenceAndBasicProperties}{}
The above choices of cofibrations, fibrations and weak equivalences equips the category $dSet$ of dendroidal sets with the structure of a [[model category]] $dSet_{CM}$. This model structure is

\begin{itemize}%
\item a left [[proper model category]]


\item a [[cofibrantly generated model category]]


\item a [[combinatorial model category]]


\item a [[symmetric monoidal category|symmetric]] [[monoidal model category]] (with respect to the \href{http://ncatlab.org/nlab/show/dendroidal%20set#BoardmanVogtTensorProduct}{Boardman-Vogt tensor product}).



\end{itemize}
\end{theorem}
This is (\hyperlink{CisinskiMoerdijk09}{Cisinski Moerdijk, theorem 2.4 and prop. 2.6}). We indicate the proof \hyperlink{ProofOfTheModelStructure}{below}.

\begin{remark}
\label{}\hypertarget{}{}
The [[cofibrantly generated model category|generating cofibrations]] $I$ are the boundary inclusion of [[trees]]

\begin{displaymath}
I = \{\partial \Omega[T] \hookrightarrow \Omega[T]\}
  \,.
\end{displaymath}
A set of generating acyclic cofibrations is guaranteed to exist, but no good explicit characterization is known to date.

\end{remark}
This is (\hyperlink{CisinskiMoerdijk09}{CisMoe, cor. 6.17}).

\begin{prop}
\label{}\hypertarget{}{}
A morphism $j : X \to Y$ between cofibrant objects in $dSet$ is a weak equivalence precisely if for all fibrant objects $A$ the morphism

\begin{displaymath}
\tau : dSet(Y,A) \to \tau dSet(X,A)
\end{displaymath}
is an [[equivalence of categories]], where $\tau : SSet \to Cat$ is the [[left adjoint]] to the [[nerve]].

\end{prop}
This is in (\hyperlink{Moerdijk}{Moerdijk, section 8.4}).

\hypertarget{ProofOfTheModelStructure}{}\paragraph*{{Proof}}\label{ProofOfTheModelStructure}

We state a list of lemmas to establish theorem \ref{TheModelStructureExistenceAndBasicProperties}.

The \textbf{strategy} is the following: we [[over topos|slice]] $dSet$ over a normal [[resolution]] $E_\infty$ (a model of the [[E-∞ operad]]) of the terminal object. Since over a normal object all morphisms are normal, we have a chance to establish a \emph{[[Cisinski model structure]]} on $dSet_{/E_\infty}$. This indeed exists, prop. \ref{ModelStructureOnSlices} below, and the model structure on $dSet$ itself can then be [[transferred model structure|transferred]] along the [[inverse image]] of the [[etale geometric morphism]] $dSet_{/E_\infty} \to dSet$, prop. \ref{TransferOfTheModelStructure}.

\begin{prop}
\label{}\hypertarget{}{}
For $A \to B$ an inner [[anodyne extension]] and $X \to Y$ a normal monomorphism of [[dendroidal sets]], the [[pushout-product axiom|pushout-product morphism]]

\begin{displaymath}
A \otimes Y \cup B \otimes X \to B\otimes Y
\end{displaymath}
(with ``$\otimes$'' the [[Boardman-Vogt tensor product]]) is an inner anodyne extension.

\end{prop}
This is (\hyperlink{CisinskiMoerdijk09}{Cis-Moer, prop. 3.1}).

\begin{defn}
\label{JFibrations}\hypertarget{JFibrations}{}
Write $J := \{0 \stackrel{\simeq}{\to} 1\}$ for the [[codiscrete groupoid]] on two objects. We use the same symbol for its image along $N_d i_! : Cat \hookrightarrow dSet$.

The \textbf{$J$-anodyne extensions} in $dSet$ are the morphisms generated under [[pushouts]], [[transfinite composition]] and [[retracts]] from the the inner [[anodyne extensions]] and from the [[pushout-product axiom|pushout products]] of $\{e\} \to J$ with the tree boundary inclusion

\begin{displaymath}
\{
    \partial \Omega[T] \otimes J \; \cup \;
    \Omega[T] \otimes \{0\}
    \to
    \Omega[T] \otimes J
  \}_{e \in \{0,1\}, T \in \Omega}
  \,.
\end{displaymath}
Call a morphism a \textbf{$J$-fibration} if it has the [[right lifting property]] against $J$-anodyne extensions.

\end{defn}
This is (\hyperlink{CisinskiMoerdijk09}{Cis-Moer, 3.2}).

\begin{prop}
\label{}\hypertarget{}{}
For $A \to B$ a $J$-anodyne extension and $X \to Y$ a normal monomorphism of [[dendroidal sets]], the [[pushout-product axiom|pushout-product morphism]]

\begin{displaymath}
A \otimes Y \cup B \otimes X \to B\otimes Y
\end{displaymath}
is a $J$-anodyne extension.

\end{prop}
This is (\hyperlink{CisinskiMoerdijk09}{Cis-Moer, prop. 3.3}).

\begin{defn}
\label{}\hypertarget{}{}
For $X \in dSet$, the BV-tensor product with

\begin{displaymath}
\{0\} \coprod \{1\} \to J \to \eta
\end{displaymath}
defines an [[interval object]]

\begin{displaymath}
X \coprod X \to J \otimes X \to X
  \,.
\end{displaymath}
This is compatible with slicing, in that for any $B \in sSet$ and $a : X \to B$ a given morphism, we have an interval object in the [[over category]] $dSet_{/B}$ given by a diagram

\begin{displaymath}
\itexarray{
    X \coprod X &\to& J \otimes X &\to& X
    \\
    & {}_{(a,a)}\searrow & \downarrow & \swarrow_{a}
    \\
    && B
  }
  \,.
\end{displaymath}
If in the above $B$ is normal and $a : X \to B$ is $J$-anodyne, then the [[left homotopies]] given by the above cylinder defines a notion of [[left homotopy]] in $dSet_{/B}$.

Say a morphism $f : A \to A'$ in $dSet_{/B}$ is a \textbf{$B$-equivalence} if it is a homotopy equivalence with respect to this notion of homotopy, hence if for all $J$-fibrations $X \to B$ the induced morphism

\begin{displaymath}
[A',X]_{\sim_B} \to [A,X]_{\sim B}
\end{displaymath}
is a bijection.

\end{defn}
\begin{prop}
\label{ModelStructureOnSlices}\hypertarget{ModelStructureOnSlices}{}
Let $B \in dSet$ be normal. Then $dSet_{/B}$ carries a [[left proper model category|left proper]] [[cofibrantly generated model category]] for which

\begin{itemize}%
\item the weak equivalences are the $B$-equivalences;


\item the cofibrations are the monomorphisms;


\item the fibrant objects are the $J$-fibrations into $B$;


\item a morphism between fibrant objects is a fibration precisely if its image in $dSet$ is a $J$-fibration.



\end{itemize}
\end{prop}
This is (\hyperlink{CisinskiMoerdijk09}{Cis-Moer, prop. 3.5}).

\begin{proof}
Notice that

\begin{enumerate}%
\item every monomorphism over a normal object $B$ is normal,


\item the [[over topos]] (see there) $PSh(\Omega)_{/B}$ may be identified with presheaves on the slice site

\begin{displaymath}
PSh(\Omega)_{/B} \simeq PSh(\Omega_{/B})
  \,.
\end{displaymath}


\end{enumerate}
Therefore for normal $B$ the slice $dSet_{/B}$, as opposed to $dSet$ itself, has a chance to carry a [[Cisinski model structure]]. And this is indeed the case: one checks that

\begin{itemize}%
\item $J \otimes (-)$ is a \emph{functorial cyclinder} in the sense of \emph{\href{http://ncatlab.org/nlab/show/Cisinski+model+structure#FunctorialCylinder}{this definition}};


\item the class of $J$-anodyne extensions is a class of corresponding anodyne extensions in the sense of \emph{\href{http://ncatlab.org/nlab/show/Cisinski+model+structure#AnodyneExtensions}{this definition}}


\item so that together these form a \emph{homotopical structure} on $dSet_{/B}$ in the sense of \emph{\href{http://ncatlab.org/nlab/show/Cisinski+model+structure#HomotopicalStructure}{this definition}}.



\end{itemize}
The statement then is a special case of \emph{\href{http://ncatlab.org/nlab/show/Cisinski+model+structure#ModelStructureFromHomotopicalStructure}{this theorem}} at \emph{[[Cisinski model structure]]}.

\end{proof}
Let $E_\infty \in dSet$ be any normal dendroidal set such that the terminal morphism $E_\infty \to *$ is an acyclic fibration in that it has the [[right lifting property]] against the tree boundary inclusions.

By the general discussion at [[over topos]] we have an [[adjunction]]

\begin{displaymath}
(p_! \dashv p^*)
  \,.
  dSet
    \stackrel{\overset{p_!}{\leftarrow}}{\underset{p^*}{\to}}
  dSet_{/E_\infty}
  \,,
\end{displaymath}
where $p_!$ simply forgets the map to $E_\infty$ and where $p^*$ forms the [[product]] with $E_\infty$

\begin{prop}
\label{TransferOfTheModelStructure}\hypertarget{TransferOfTheModelStructure}{}
The \emph{[[transferred model structure]]} on $dSet$ along the [[right adjoint]] $p^*$ of the model structure from prop. \ref{ModelStructureOnSlices} exists.

This is the model structure characterized in theorem \ref{TheModelStructureExistenceAndBasicProperties}.

\end{prop}
This appears as (\hyperlink{CisinskiMoerdijk09}{Cis-Moer, prop. 3.12}).

So far this establishes the existence of the model structure and that every dendroidal inner Kan complex is fibrant. Below in \emph{\hyperlink{CharacterizationOfTheFibrantObjects}{characterization of the fibrant objects}} we consider the converse statement: that the fibrant objects are precisely the inner Kan complexes.

\hypertarget{CharacterizationOfTheFibrantObjects}{}\paragraph*{{Characterization of the fibrations}}\label{CharacterizationOfTheFibrantObjects}

\begin{prop}
\label{}\hypertarget{}{}
A dendroidal set is $J$-fibrant, def. \ref{JFibrations}, hence fibrant in $dSet_{CM}$, precisely if it is an inner Kan complex.

A morphism $f : X \to Y$ in $dSet$ between inner Kan complexes is a $J$-fibration, hence a fibration in $dSet_{CM}$, precisely if

\begin{enumerate}%
\item it is an [[inner Kan fibration]];


\item on [[homotopy categories]] $\tau i^* f$ is an [[isofibration]].



\end{enumerate}
\end{prop}
This is (\hyperlink{CisinskiMoerdijk09}{Cisinski Moerdijk, theorem 5.10}).

\hypertarget{MonoidalModelCategoryStructure}{}\subsubsection*{{Monoidal model category structure}}\label{MonoidalModelCategoryStructure}

\begin{prop}
\label{}\hypertarget{}{}
With respect to the [[Boardman-Vogt tensor product]] on [[dendroidal sets]], the model structure $dSet_{CM}$ is a [[symmetric monoidal category|symmetric]] [[monoidal model category]].

\end{prop}
This is (\hyperlink{CisinskiMoerdijk09}{Cis-Moer, prop. 3.17}).

\begin{cor}
\label{EnrichmentOverItself}\hypertarget{EnrichmentOverItself}{}
With respect to the [[internal hom]] corresponding to the [[Boardman-Vogt tensor product]], $dSet_{CM}$ is a $dSet_{CM}$-[[enriched model category]].

\end{cor}
\hypertarget{Enrichment}{}\subsubsection*{{Other enrichments of the underlying category}}\label{Enrichment}

\begin{prop}
\label{}\hypertarget{}{}
\textbf{(compatibility with the Joyal model structure)}

Let $|$ be the tree with a single leaf and no vertex. Then the [[overcategory]] $dSet/\Omega[|]$ is canonically isomorphic to [[sSet]].

The model structure on [[sSet]] induced this way as the [[model structure on an overcategory]] from the model structure on $dSet$ coincides with the [[model structure for quasi-categories]].

\end{prop}
This is for instance (\hyperlink{Moerdijk}{Moerdijk, proposition 8.4.3}).

\begin{prop}
\label{}\hypertarget{}{}
This model category is naturally an $sSet_{Joyal}$-[[enriched model category]], where $sSet_{Joyal}$ is the [[model structure for quasi-categories]].

\end{prop}
\begin{proof}
This follows from the fact, cor. \ref{EnrichmentOverItself}, that $dSet_{CM}$ is a [[monoidal model category]] and the fact that the functor $i^*: dSet \to sSet_{Joyal}$ is a [[right Quillen functor]].

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
However, $dSet_{CM}$ is \emph{not} an [[enriched model category]] over $sSet_{Quillen}$, the \emph{standard} [[model structure on simplicial sets]] (but see \emph{[[model structure for dendroidal Cartesian fibrations]]}). But it comes close, as the following propositions show.

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
Write

\begin{displaymath}
\mathcal{Hom} : dSet^{op} \times dSet \to dSet
\end{displaymath}
for the [[internal hom]] corresponding to the [[Boardman-Vogt tensor product]].

For $A$ normal and $X$ an inner Kan dendroidal set, write

\begin{displaymath}
\mathcal{hom}(A,X) := i^* \mathcal{Hom}(A,X)
\end{displaymath}
for the underlying [[quasi-category]], and write

\begin{displaymath}
k(A,X) := Core(\mathcal{hom}(A,X)) \in KanCplx
\end{displaymath}
for the maximal [[Kan complex]] inside the [[quasi-category]] inside the [[internal hom]].

Write

\begin{displaymath}
-^{(-)} : sSet^{op} \times dSet \to dSet
\end{displaymath}
for the corresponding [[powering]], characterized by the existence of a [[natural isomorphism]]

\begin{displaymath}
Hom_{sSet}(K, k(A,X)) \simeq Hom_{dSet}(A, X^{(K)})
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
For $p : X \to Y$ a fibration between fibrant dendroidal sets (hence an [[inner Kan fibration]] and an [[isofibration]] on the underlying [[homotopy category]]), and for $A \to B$ a normal monomorphism, the induced morphism

\begin{displaymath}
k(B,X) \to k(B,Y) \times_{k(A,Y)} k(A,X)
\end{displaymath}
is a [[Kan fibration]] between [[Kan complexes]].

\end{prop}
This is (\hyperlink{CisinskiMoerdijk09}{Cis-Moer, prop. 6.7}).

\begin{prop}
\label{}\hypertarget{}{}
If $A \to B$ is the above is an [[anodyne extension]] (acyclic monomorphism) of simplicial sets, then

\begin{displaymath}
X^{(B)} \to Y^{(B)} \times_{Y^{(A)}} X^{(A)}
\end{displaymath}
is an acyclic fibration in $dSet_{CM}$.

\end{prop}
This is (\hyperlink{CisinskiMoerdijk09}{Cis-Moer, cor. 6.9}).

\begin{prop}
\label{}\hypertarget{}{}
For $A$ normal and $X$ fibrant, the Kan complex

\begin{displaymath}
k(A,X)
  \simeq
  \mathbb{R}Hom(A,X)
\end{displaymath}
is the correct [[derived hom-space]] of $dSet_{CM}$.

\end{prop}
\begin{proof}
One checks that $n \mapsto X^{(\Delta[n])}$ is a fibrant resolution of $X$ in the [[Reedy model structure]] $[\Delta^{op}, dSet_{CM}]_{Reedy}$. By the discussion at \emph{[[simplicial model category]]} and \emph{[[derived hom-space]]} the latter is therefore given by the simplicial set

\begin{displaymath}
n \mapsto Hom_{dSet}(A, X^{(\Delta[n])})
  \,.
\end{displaymath}
By the [[tensoring]]-definition of $X^{(\Delta[n])}$ this is isomorphic to

\begin{displaymath}
\cdots = Hom_{sSet}(\Delta[n], k(A,X)) = k(A,X)_n
  \,.
\end{displaymath}
\end{proof}
\hypertarget{relation_to_other_model_structures}{}\subsubsection*{{Relation to other model structures}}\label{relation_to_other_model_structures}

We discuss the relation of the model structure on dendroidal sets to other model category structures for operads.

See the \emph{[[table - models for (infinity,1)-operads]]} for an overview.

\hypertarget{model_structure_for_quasicategories}{}\paragraph*{{Model structure for quasi-categories}}\label{model_structure_for_quasicategories}

\begin{proposition}
\label{}\hypertarget{}{}
The [[adjunction]]

\begin{displaymath}
(j_! \dashv j^*) : 
   dSet
    \stackrel{\overset{j_!}{\leftarrow}}{\underset{j^*}{\to}}
   sSet
\end{displaymath}
induced from the inclusion $j : \Delta \hookrightarrow \Omega$ constitutes a [[Quillen adjunction]] between the above model structure on dendroidal sets, and the [[model structure for quasi-categories]].

\end{proposition}
\begin{proof}
By the proof of (\hyperlink{CisinskiMoerdijk09}{Cisinski-Moerdijk, cor. 2.10}) the model structure for quasi-categories is in fact the restriction, along $j_!$, of the model structure on dendroidal sets. Therefore $j_!$ is left Quillen.

\end{proof}
\hypertarget{RelationToModelStructureForDendroidalCompleteSegal}{}\paragraph*{{Model structure for dendroidal complete Segal spaces}}\label{RelationToModelStructureForDendroidalCompleteSegal}

There is a [[Quillen equivalence]] to the [[model structure for dendroidal complete Segal spaces]] (see there). A crucial step in the proof is the following expression of the acyclic cofibrations on $dSet_{CM}$ in terms of the dendroidal interval $J_d$ as follows.

\begin{prop}
\label{}\hypertarget{}{}
The class of acyclic cofibrations between normal dendroidal sets is the smallest class of morphisms between normal dendroidal sets

\begin{itemize}%
\item which contains the $J$-anodyne extensions;


\item with left cancellation property: if a composite $\stackrel{i}{\to} \stackrel{j}{\to}$ is in the class and $i$ is, then so is $j$.



\end{itemize}
\end{prop}
(\hyperlink{CisinskiMoerdijk09}{Cis-Moer 09, prop. 3.16})

\hypertarget{RelationToModelStrictureOnSimplicialOperads}{}\paragraph*{{Model structure on simplicial operads}}\label{RelationToModelStrictureOnSimplicialOperads}

There exists also a [[model structure on simplicial operads]], which is [[Quillen equivalence|Quillen equivalent]] to the model structure on dendroidal sets.

This Quillen equivalence is an operadic generalization of the Quillen equivalence between the [[model structure on sSet-categories]] and the [[model structure for quasi-categories]].

\begin{theorem}
\label{}\hypertarget{}{}
There exists an [[adjunction]]

\begin{displaymath}
(W_! \dashv hcN_d) 
   : 
  sSet Operad 
    \stackrel{\overset{W_!}{\leftarrow}}{\underset{hcN_d}{\to}}
  dSet 
  \,,
\end{displaymath}
Whise [[right adjoint]] is the [[dendroidal homotopy coherent nerve]].

This is a [[Quillen equivalence]] between the model structure on dendroidal sets, and the [[model structure on simplicial operads]].

\end{theorem}
This is (\hyperlink{CisinskiMoerdijk11}{Cisinski-Moerdijk 11, theorem 815}).

\begin{remark}
\label{}\hypertarget{}{}
Under the inclusions (see the discussion at \emph{[[dendroidal set]]})

\begin{displaymath}
\itexarray{
     sSet Cat &\hookrightarrow & sSet Operad
     \\
     && {}^{\mathllap{hcN_d}}\downarrow \uparrow^{\mathrlap{W_!}}
     \\
     sSet \simeq dSet/\eta &\hookrightarrow & dSet
  }
\end{displaymath}
this restricts to the Quillen equivalence between the [[model structure on sSet-categories]] and the [[model structure for quasi-categories]] discussed at \emph{[[relation between quasi-categories and simplicial categories]]}.

\end{remark}
\hypertarget{model_structure_on_operads}{}\paragraph*{{Model structure on $Set$-Operads}}\label{model_structure_on_operads}

\begin{remark}
\label{}\hypertarget{}{}
Write [[Operad]] for the category of [[symmetric operads]] over [[Set]].

The [[dendroidal set|dendroidal nerve adjunction]]

\begin{displaymath}
(\tau_d \dashv N_d) : 
  Operad
   \stackrel{\overset{\tau_d}{\leftarrow}}{\underset{N_d}{\to}}
  dSet
\end{displaymath}
is a [[Quillen adjunction]] between the model structure on dendroidal sets, def. \ref{ModelStructureOnDendroidalSets}, and the [[canonical model structure on Operad]].

Moreover, $N_d$ detects and preserves weak equivalences, while $\tau_d$ preserves weak equivalences.

\end{remark}
This is (\hyperlink{CisinskiMoerdijk09}{Cisinski-Moerdijk 09, prop. 2.5}).

\hypertarget{model_structure_for_symmetric_monoidal_categories}{}\paragraph*{{Model structure for symmetric monoidal $(\infty,1)$-categories}}\label{model_structure_for_symmetric_monoidal_categories}

\begin{prop}
\label{}\hypertarget{}{}
There is a [[Quillen adjunction]]

\begin{displaymath}
dSet_{CM}
   \stackrel{\overset{id}{\leftarrow}}{\underset{id}{\to}}
  dSet_{He}
\end{displaymath}
which exhibits the [[model structure for dendroidal left fibrations]] as a [[Bousfield localization of model categories|left Bousfield localization]] of the Cisinski-Moerdijk model structure on dendroidal sets.

\end{prop}
See (\hyperlink{Heuts}{Heuts, remark 6.8.0.2}).

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

\begin{itemize}%
\item [[canonical model structure on Operad]],


\item [[model structure on operads]]


\item [[model structure for dendroidal complete Segal spaces]]


\item [[table - models for (infinity,1)-operads]]



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

A useful discussion of of the model structure on dendroidal sets is section 8 of

\begin{itemize}%
\item [[Ieke Moerdijk]], \emph{Lectures on dendroidal sets} , lectures given at the Barcelona workshop on \emph{\href{http://www.crm.es/HigherCategories/}{Simplicial methods in higher categories}} (2008) (\href{http://www.crm.es/Publications/quaderns/Quadern45-1.pdf}{preliminary writeup})

\end{itemize}
An expanded and polished version has meanwhile been written up by Javier Guiti\'e{}rrez and should appear in print soon. An electronic copy is probably available on request.

The model structure was originally given in

\begin{itemize}%
\item [[Denis-Charles Cisinski]], [[Ieke Moerdijk]], \emph{Dendroidal sets as models for homotopy operads} (\href{http://arxiv.org/abs/0902.1954}{arXiv:0902.1954})

\end{itemize}
making heavy use of results on [[Cisinski model structures]] from

\begin{itemize}%
\item [[Denis-Charles Cisinski]], \emph{[[joyalscatlab:Les préfaisceaux comme type d'homotopie]]}, Ast\'e{}risque, Volume 308, Soc. Math. France (2006), 392 pages (\href{http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf}{pdf})

\end{itemize}
A detailed discussion of the fibrant objects in the model structure is in

\begin{itemize}%
\item [[Ieke Moerdijk]] [[Ittay Weiss]], \emph{On inner Kan complexes in the category of dendroidal sets} (\href{http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6W9F-4VM2KC8-1&_user=457046&_rdoc=1&_fmt=&_orig=search&_sort=d&_docanchor=&view=c&_searchStrId=1115574112&_rerunOrigin=google&_acct=C000021878&_version=1&_urlVersion=0&_userid=457046&md5=5eb2307e02ed1aa9e6fe2c7809346546}{web})

\end{itemize}
The proof of the Quillen equivalence between the model structure on dendroidal sets and that on $sSet$-operads is given in

\begin{itemize}%
\item [[Denis-Charles Cisinski]], [[Ieke Moerdijk]], \emph{Dendroidal sets and simplicial operads} (\href{http://arxiv.org/abs/1109.1004}{arXiv:1109.1004})

\end{itemize}
The relation to the [[model structure for dendroidal Cartesian fibrations]] and the [[model structure for dendroidal left fibrations]] is discussed in

\begin{itemize}%
\item [[Gijs Heuts]], \emph{Algebras over infinity-Operads} (\href{http://arxiv.org/abs/1110.1776}{arXiv:1110.1776})

\end{itemize}


\end{document}