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\begin{document}

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\section*{model structure for Segal operads}

\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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{segal_operads}{Segal operads}\dotfill \pageref*{segal_operads} \linebreak
\noindent\hyperlink{special_morphisms}{Special morphisms}\dotfill \pageref*{special_morphisms} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{of_the_various_classes_of_morphisms}{Of the various classes of morphisms}\dotfill \pageref*{of_the_various_classes_of_morphisms} \linebreak
\noindent\hyperlink{of_the_model_structure_itself}{Of the model structure itself}\dotfill \pageref*{of_the_model_structure_itself} \linebreak
\noindent\hyperlink{relation_to_other_model_structures}{Relation to other model structures}\dotfill \pageref*{relation_to_other_model_structures} \linebreak
\noindent\hyperlink{to_dendroidal_complete_segal_spaces}{To dendroidal complete Segal spaces}\dotfill \pageref*{to_dendroidal_complete_segal_spaces} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The \emph{model structure for Segal operads} is a [[presentable (∞,1)-category|presentation]] of the [[(∞,1)-category]] of [[(∞,1)-operads]] regarding these as [[enriched (∞,1)-category|∞Grpd-enriched]] [[operads]].

It is the [[operad|operadic]] analog of the \emph{[[model structure for Segal categories]]}: its [[fibrant objects]] are operadic analogs of [[Segal categories]].

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

Write $\Omega$ for the [[tree category]], the [[site]] for [[dendroidal sets]].

\hypertarget{segal_operads}{}\subsubsection*{{Segal operads}}\label{segal_operads}

Write $\eta$ for the tree with a single edge and no vertices. Write

\begin{displaymath}
sdSet := [\Omega^{op}, sSet]
\end{displaymath}
for the category of [[simplicial presheaves]] on the tree category -- \emph{simplicial dendroidal sets} or \emph{dendroidal simplicial sets} (see [[model structure for complete dendroidal Segal spaces]] for more on this).

\begin{defn}
\label{InclusionOfSegalPreOperads}\hypertarget{InclusionOfSegalPreOperads}{}
A \textbf{Segal pre-operad} $X \in [\Omega^{op}, sSet]$ is a simplicial dendroidal set such that $X(\eta)$ is a [[discrete object|discrete]] [[simplicial set]] (a plain set regarded as a simplicially constant simplicial set). Write

\begin{displaymath}
SegalPreOperad \hookrightarrow [\Omega^{op}, sSet]
\end{displaymath}
for the [[full subcategory]] on the Segal pre-operads.

A \textbf{Segal operad} is a Segal pre-operad such that for every [[tree]] $T \in \Omega$ the [[powering]]

\begin{displaymath}
X^{\Omega[T]} \to X^{Sp(T)}
  \in
  sSet
\end{displaymath}
of the [[spine]] inclusion $(Sp(T) \hookrightarrow T) \in$ [[dendroidal set|dSet]] into $X$ is an [[weak equivalence|acyclic]] [[Kan fibration]]. Write

\begin{displaymath}
SegalOperad \hookrightarrow SegalPreOperad
\end{displaymath}
for the [[full subcategory]] on the Segal operads.

A \textbf{Reedy-fibrant Segal operad} is a Segal operad which is moreover fibrant in the [[generalized Reedy model structure]] $[\Omega^{op}, sSet]_{gReedy}$.

\end{defn}
This is (\hyperlink{CisinskiMoerdijk}{Cisinski-Moerdijk, def. 7.1, def. 8.1}).

\begin{remark}
\label{}\hypertarget{}{}
The definition of Segal pre-operads encodes a [[set]] of colors of an [[operad]], together with for each [[tree]] $T$ an [[∞-groupoid]] of operations in the operad of the shape of this tree --- notably $\infty$-groupoids of $n$-ary operations if the tree is the $n$-corolla, $T = C_n$.

The condition on Segal operads encodes the existence of \emph{composition} of these operad operations by [[∞-anafunctors]]. See the discussion at \emph{[[Segal category]]} for more on this.

The Reedy fibrancy condition is mostly a technical convenience.

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
The inclusion def. \ref{InclusionOfSegalPreOperads} has a [[left adjoint|left]] and [[right adjoint|right]] [[adjoint functors]]

\begin{displaymath}
sdSet 
  \stackrel{\overset{\gamma_!}{\to}}{\stackrel{\overset{\gamma^*}{\leftarrow}}{\underset{\gamma_*}{\to}}}
  SegalPreOperad
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
One way to see the existence of the adjoints is to note that $SegalPreOperad$ is a [[category of presheaves]] over the [[site]] $S(\Omega)$ which is the [[localization]] of $\Omega \times \Delta$ at morphisms of the form $(-,Id_\eta)$, where $\eta$ is the tree with one edge and no vertex. Write

\begin{displaymath}
\gamma : \Delta \times \Omega \to S(\Omega)
\end{displaymath}
for the [[localization]] functor, then the inclusion of Segal pre-operads is the precomposition with this functor

\begin{displaymath}
\gamma^* : SegalPreOperad \simeq [S(\Omega)^{op}, sSet] \hookrightarrow [\Omega^{op}, sSet]
  \,.
\end{displaymath}
Therefore the left and right adjoint to $\gamma^*$ are given by left and right [[Kan extension]] along $\gamma$.

Explicitly, these adjoints are given as follows.

For $X \in [\Omega^{op}, sSet]$, the Segal pre-operad $\gamma_!(X)$ sends a tree $T$ either to $X(T)$, if $T$ is non-linear, hence if it admits no morphism to $\eta$, or else to the [[pushout]]

\begin{displaymath}
\itexarray{
    X(\eta) &\to& X(T)
    \\
    \downarrow && \downarrow
    \\
    \pi_0 X(\eta) &\to& \gamma_!(X)(T)
  }
\end{displaymath}
in [[sSet]], where the top morphism is $X(T \to \eta)$ for the unique morphism to $\eta$.

In words, $\gamma_!(X)$ is obtained from $X$ precisely by contracting the simplicial set of colors to its set of connected components.

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

We discuss morphisms between Segal pre-operads with special properties, which will appear in the model structure.

\begin{defn}
\label{NormalMonomorphism}\hypertarget{NormalMonomorphism}{}
Say a morphism $f$ in $SegalPreOperad$ is a \textbf{normal monomorphism} precisely if $\gamma^*(f)$ is a normal monomorphism (see [[generalized Reedy model structure]]), which in turn is the case if it is simplicial-degreewise a normal morphisms of [[dendroidal sets]] (see there for details).

Correspondingly, a Segal pro-operad $X$ is called \emph{normal} if $\emptyset \to X$ is a normal monomorphism.

\end{defn}
\begin{defn}
\label{PreOperadAcyclicFibration}\hypertarget{PreOperadAcyclicFibration}{}
A morphism in $SegalPreOperad$ is called an \textbf{acyclic fibration} precisely if it has the [[right lifting property]] against all normal monomorphisms, def. \ref{NormalMonomorphism}.

\end{defn}
\begin{defn}
\label{SegalWeakEquivalence}\hypertarget{SegalWeakEquivalence}{}
Say a morphism $f$ in $SegalPreOperad$ is a \emph{Segal weak equivalence} precisely if $\gamma^*(f)$ is a weak equivalence in the [[model structure for dendroidal complete Segal spaces]] $[\Omega^{op}m, sSet]_{gReedy \atop cSegal}$.

\end{defn}
\begin{defn}
\label{WeakEquivalencesAndFibrations}\hypertarget{WeakEquivalencesAndFibrations}{}
Call a morphism in $SegalPreOperad$

\begin{itemize}%
\item a weak equivalence precisely if it is a Segal weak equivalence, def. \ref{SegalWeakEquivalence};


\item a cofibration precisely if it is a normal monomorphism, def. \ref{NormalMonomorphism}.



\end{itemize}
\end{defn}
Theorem \ref{ExistenceOfTheModelStructure} below asserts that this is indeed a model category struture whose fibrant objects are the Segal operads.

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

\hypertarget{of_the_various_classes_of_morphisms}{}\subsubsection*{{Of the various classes of morphisms}}\label{of_the_various_classes_of_morphisms}

\begin{lemma}
\label{}\hypertarget{}{}
If $f : X \to Y$ in $[\Omega^{op}, sSet]$ is a normal monomorphism and $\pi_0 X(\eta) \to \pi_0 Y(\eta)$ is a monomorphism, then $\gamma_!(f)$ is normal in $SegalPreOperad$.

\end{lemma}
(\hyperlink{CisinskiMoerdijk}{Cis-Moer, lemma 7.4}).

\begin{prop}
\label{}\hypertarget{}{}
The class of normal monomorphisms in $SegalPreOperad$ is [[cofibrantly generated model category|generated]] (under [[pushout]], [[transfinite composition]] and [[retracts]]) by the set

\begin{displaymath}
\{
     \gamma_!(\partial \Delta[n] \times \Omega[T]  \cup \Delta[n] \times \partial \Omega[T])
     \to
    \gamma_! (\Delta[n], \Omega[T])
  \}_{n \in \Delta, T \in \Omega, {\vert T\vert} \geq 1}
 \cup
  \{
    \emptyset \to \eta
  \}
\end{displaymath}
\end{prop}
(\hyperlink{CisinskiMoerdijk}{Cis-Moer, prop 7.5}).

\begin{prop}
\label{}\hypertarget{}{}
Let $X \in [\Omega^{op}, sSet]_{gReedy \atop Segal}$ be fibrant. Then $\gamma_* X$ is a Reedy fibrant Segal operad. If $X$ is moreover fibrant in $[\Omega^{op}, sSet]_{gReedy \atop cSegal}$ then the [[unit of an adjunction|counit]] $\gamma^* \gamma* X \to X$ is a weak equivalence in $[\Omega^{op}, sSet]_{gReedy \atop cSegal}$.

\end{prop}
(\hyperlink{CisinskiMoerdijk}{Cis-Moer, prop 8.2}).

\begin{lemma}
\label{AcyclicFibrationsAreIndeedWeakEquivalences}\hypertarget{AcyclicFibrationsAreIndeedWeakEquivalences}{}
An acyclic fibration in $SegalPreOperad$, def. \ref{PreOperadAcyclicFibration}, is also a weak equivalence in $[\Omega^{op}, sSet]_{gReedy \atop Segal}$.

\end{lemma}
(\hyperlink{CisinskiMoerdijk}{Cis-Moer, prop 8.12}).

\hypertarget{of_the_model_structure_itself}{}\subsubsection*{{Of the model structure itself}}\label{of_the_model_structure_itself}

\begin{theorem}
\label{ExistenceOfTheModelStructure}\hypertarget{ExistenceOfTheModelStructure}{}
The structures in def. \ref{WeakEquivalencesAndFibrations} make the category $SegalPreOperad$ a [[model category]] which is

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


\item [[left proper model category|left proper]].



\end{itemize}
\end{theorem}
This is (\hyperlink{CisinskiMoerdijk}{Cis-Moer, theorem 8.13}).

\begin{proof}
The existence of the cofibrantly generated model structure follows with \emph{\href{combinatorial+model+category#SmithTheorem}{Smith's theorem}}: by the discussion there it is sufficient to notice that

\begin{enumerate}%
\item the Segal equivalences are an accessibly embedded accessible full subcategory of the arrow category;


\item the acyclic cofibrations are closed under pushout and retract;

(both of these because these morphisms come from the [[combinatorial model category]] $[\Omega^{op}, sSet]_{gReedy \atop cSegal}$)


\item the morphisms with right lifting against the normal monomorphisms are weak equivalences, by lemma \ref{AcyclicFibrationsAreIndeedWeakEquivalences}.



\end{enumerate}
\end{proof}
\hypertarget{relation_to_other_model_structures}{}\subsubsection*{{Relation to other model structures}}\label{relation_to_other_model_structures}

We discuss the relation to various other model structures for operads. For an overview see \emph{[[table - models for (infinity,1)-operads]]}.

\hypertarget{to_dendroidal_complete_segal_spaces}{}\paragraph*{{To dendroidal complete Segal spaces}}\label{to_dendroidal_complete_segal_spaces}

(\ldots{}) [[model structure for dendroidal complete Segal spaces]]

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

\begin{itemize}%
\item [[Denis-Charles Cisinski]], [[Ieke Moerdijk]], \emph{Dendroidal Segal spaces and infinity-operads} (\href{http://arxiv.org/abs/1010.4956}{arXiv:1010.4956})

\end{itemize}
[[!redirects Segal operad]] [[!redirects Segal operads]]



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