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\begin{document}

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\section*{dendroidal homotopy coherent nerve}

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

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - 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{BVResolutionOfTrees}{BV resolution of trees}\dotfill \pageref*{BVResolutionOfTrees} \linebreak
\noindent\hyperlink{the_homotopy_coherent_nerve}{The homotopy coherent nerve}\dotfill \pageref*{the_homotopy_coherent_nerve} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{specialization_to_categories}{Specialization to categories}\dotfill \pageref*{specialization_to_categories} \linebreak
\noindent\hyperlink{DendroidalInnerKanComplexes}{Dendroidal inner Kan complexes}\dotfill \pageref*{DendroidalInnerKanComplexes} \linebreak
\noindent\hyperlink{LeftAdjoint}{Left adjoint}\dotfill \pageref*{LeftAdjoint} \linebreak
\noindent\hyperlink{relation_to_ordinary_dendroidal_nerve}{Relation to ordinary dendroidal nerve}\dotfill \pageref*{relation_to_ordinary_dendroidal_nerve} \linebreak
\noindent\hyperlink{quillen_equivalence}{Quillen equivalence}\dotfill \pageref*{quillen_equivalence} \linebreak
\noindent\hyperlink{resolution_and_rectification}{Resolution and rectification}\dotfill \pageref*{resolution_and_rectification} \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 \emph{dendroidal homotopy coherent nerve} is an [[operad|operadic]] generalization of the standard [[homotopy coherent nerve]]. It is a [[functor]]

\begin{displaymath}
hcN_d : Top Operad \to dSet
\end{displaymath}
from the [[category]] of [[Top]]-[[operads]] to that of [[dendroidal sets]], given by

\begin{displaymath}
hcsN_d(P) : T \mapsto dSet(W_H(T), P)
  \,,
\end{displaymath}
where $T$ is an object of the [[tree category]], regarded as a free [[symmetric operad]], and $W_H(T)$ is its [[Boardman-Vogt resolution]].

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

Throughout, let $\mathcal{E}$ be a [[symmetric monoidal category|symmetric]] [[monoidal model category]] equipped with an [[interval object]] $H$ as discussed at \emph{[[model structure on operads]]} and at \emph{[[Boardman-Vogt resolution]]}. We consider multi-coloured [[symmetric operads]] ([[symmetric multicategories]]) enriched in $\mathcal{E}$.

Standard examples are $\mathcal{E} =$ [[Top]], [[sSet]], which yields [[topological operads]] and [[simplicial operads]], respectively.

\hypertarget{BVResolutionOfTrees}{}\subsubsection*{{BV resolution of trees}}\label{BVResolutionOfTrees}

We discuss in detail what the [[Boardman-Vogt resolution]] of [[operads]] free on an object in the [[tree category]] $\Omega$ is like (see \emph{[[dendroidal set]]} for details on trees as operads).

Let

\begin{displaymath}
Symm : \mathcal{E} Operad_{planar} \to \mathcal{E} Operad
\end{displaymath}
be the symmetrization functor, the [[left adjoint]] to the [[forgetful functor]] from [[symmetric operads]] to [[planar operads]].

\begin{prop}
\label{}\hypertarget{}{}
The BV resolution commutes with symmetrization: if $T = Symm(\bar T)$, then

\begin{displaymath}
W(T) = Symm(W(\bar T))
  \,.
\end{displaymath}
\end{prop}
Therefore we describe in the following explicitly the BV-resolution of planar trees, that of non-planar trees then being the symmetrization of that construction.

\begin{prop}
\label{ComponentDescriptionofBVofTrees}\hypertarget{ComponentDescriptionofBVofTrees}{}
For $T \in \Omega_{planar}$, and $(e_1, \cdots, e_n; e)$ a tuple of colours (edges) of $T$, notice that the set of operations $T(e_1, \cdots, e_n, e)$ is the set of those subtrees $V \subset T$ such that $\{e_1, \cdots, e_n\}$ is the set of leaves and $e$ is the root of $V$.

First regard $T$ as a [[topological operad]] (with a [[discrete space]] of operations in each degree). The corresponding [[Boardman-Vogt resolution]] $W(T)$ of $T$ is the topological operad whose [[topological space]] of operations $W(T)(e_1, \cdots, e_n; e)$ is the space of \emph{labeled trees} as follows.

A point is a set of \emph{lengths} $\ell(e) \in [0,1]$, one for each inner edge $e \in I(T)$ of $T$. (\ldots{})

Hence

\begin{displaymath}
W(T)(e_1, \cdots, e_n; e) 
    \simeq 
  \coprod_{V \in T(e_1, \cdots, e_n; e)}
  (\Delta^1)^{\times i(V)}
\end{displaymath}
where the [[coproduct]] ranges over the set of subtrees $V$, as just discussed (which therefore is either the singleton set or is empty), and where $i(V)$ is the set of inner edges of $V$.

Regard then $T$ as a [[simplicial operad]]. The corresponding [[Boardman-Vogt resolution]] $W(T)$ of $T$ is the simplicial operad whose [[simplicial sets]] of operations are

\begin{displaymath}
W(T)(e_1, \cdots, e_n; e)
  =
  \coprod_{V \in T(e_1, \cdots, e_n; e)}
  \Delta[1]^{\times i(V)}
  \,.
\end{displaymath}
In general, when $T$ is regarded as an $\mathcal{E}$-operad, we have

\begin{displaymath}
W(T)(e_1, \cdots, e_n; e)
  =
  \coprod_{V \in T(e_1, \cdots, e_n; e)}
  H^{\otimes i(V)}
  \,,
\end{displaymath}
where $H$ is the given [[interval object]].

\end{prop}
\begin{prop}
\label{CompositioninWT}\hypertarget{CompositioninWT}{}
The composition operations in $W(T)$

\begin{displaymath}
\itexarray{
    W(T)(e_1, \cdots, e_n; e_0) \otimes W(T)(f_1, \cdots, f_k; e_i)
    \\
    \downarrow^{\mathrlap{\circ_i}}
    \\
    W(T)(e_1, \cdots, e_{i-1}, f_1, \cdots, f_k, e_{i+1}, \cdots, e_n)
  }
\end{displaymath}
correspond to grafting of trees $T_\sigma, T_\rho \subset T$ and ``assigning unit length to the new inner edge''. On the components as discussed above it is given by

\begin{displaymath}
\itexarray{
    H^{\otimes i(T_\sigma)} \otimes H^{\otimes i(T_\rho)}
    &&
    H^{\otimes i(T_\sigma \circ_i T_\rho)}
    \\
    \downarrow^{\mathrlap{\simeq}}
    &&
    \uparrow^{\mathrlap{\simeq}}
    \\
    H^{\otimes i(T_\sigma) \cup i(T_\rho)} \otimes I
     &\stackrel{id \otimes i_1}{\to}&
    H^{\otimes (i(T_\sigma) \cup i(T_\rho))} \otimes H
  }
\end{displaymath}
\end{prop}
\begin{prop}
\label{BVResolutionAsFunctorOnOmega}\hypertarget{BVResolutionAsFunctorOnOmega}{}
The BV-resolution of trees extends to a [[functor]] on the category of [[simplicial operad]]

\begin{displaymath}
W : \Omega \to sSet Operad
\end{displaymath}
as follows

\begin{itemize}%
\item on an inner face map $\delta_e : \partial^e\Omega[T] \to \Omega[T]$ the component of $W(\delta)$ on a subtree $V$ of $T$ that contains the edge $e$ is the product of the inclusion

\begin{displaymath}
i_0 : I \to H
\end{displaymath}
with the identity on $H^{i(V)-\{e\}}$

(meaning: if the label of an inner edge in a tree is 0, then the operations that it connects may be composed);


\item on a degenracy map $\sigma$ that sends two given unary vertices to a single one, the component of $W(\sigma)$ on subtrees containing these removes one of the factors $H$ by the map

\begin{displaymath}
H \otimes H \to H
\end{displaymath}
given by the [[interval object]] $H$. For both [[simplicial operads]] and [[topological operads]] this may be taken to be the map

\begin{displaymath}
max : \Delta^1 \times \Delta^1 \to \Delta^1
\end{displaymath}
that sends $(x,y)$ to $max(x,y)$.



\end{itemize}
\end{prop}
This is discussed in section 4.2 of (\hyperlink{CisinskiMoerdijk}{Cisinski-Moerdijk}).

\hypertarget{the_homotopy_coherent_nerve}{}\subsubsection*{{The homotopy coherent nerve}}\label{the_homotopy_coherent_nerve}

By the general discussion at \emph{[[nerve and realization]]}, the functor

\begin{displaymath}
W
  : 
  \Omega 
  \to  
  sSet Operad
\end{displaymath}
from prop. \ref{BVResolutionAsFunctorOnOmega} induces a [[nerve]] functor as follows.

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{dendroidal homotopy coherent nerve} functor is the [[functor]]

\begin{displaymath}
hcN_d : sSet Cat \to dSet
\end{displaymath}
given by

\begin{displaymath}
P \mapsto ( T \mapsto sSet Operad(W(T), P) )
  \,.
\end{displaymath}
Its [[left adjoint]] (the corresponding ``[[geometric realization]]'') we denote

\begin{displaymath}
W_! : dSet \to sSet Operad
  \,.
\end{displaymath}
\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{specialization_to_categories}{}\subsubsection*{{Specialization to categories}}\label{specialization_to_categories}

\begin{prop}
\label{}\hypertarget{}{}
When restricted to $\mathcal{E}$-[[enriched categories]], the dendroidal homotopy coherent nerve reproduces the [[homotopy coherent nerve]] of enriched categories

\begin{displaymath}
\itexarray{
    \mathcal{E} Cat &\hookrightarrow& \mathcal{E} Operad
    \\
    {}^{\mathllap{hcN}}\downarrow && \downarrow^{\mathrlap{hcN_d}}
    \\
    sSet &\hookrightarrow& dSet
  }
  \,.
\end{displaymath}
In particular for $\mathcal{E} =$ [[Top]] / [[sSet]] it reproduces the original definition of homotopy coherent nerve.

\end{prop}
\hypertarget{DendroidalInnerKanComplexes}{}\subsubsection*{{Dendroidal inner Kan complexes}}\label{DendroidalInnerKanComplexes}

\begin{prop}
\label{hcNOfLocllyFibrantOperadIsFibrant}\hypertarget{hcNOfLocllyFibrantOperadIsFibrant}{}
Let $P \in \mathcal{E} Operad$ be such that each object of operations is fibrant in $\mathcal{E}$. Then its homotopy coherent nerve $hcN_d(P)$ is a [[model structure on dendroidal sets|dendroidal inner Kan complex]].

\end{prop}
This is (\hyperlink{MoerdijkWeiss}{Moerdijk-Weiss, theorem 7.1}). This statement will also follow as a corollary from prop. \ref{WIsLeftQuillen} below.

\begin{proof}
Consider a tree $T$ and an inner edge $e$ of it. For each morphism $\phi : \Lambda^e[T] \to X$ we need to find a filler $\psi$ in

\begin{displaymath}
\itexarray{
    \Lambda^e[T] &\to& hcN_d(X)
    \\
    \downarrow & \nearrow_{\mathrlap{\psi}}
    \\
    \Omega[T]
  }
  \,.
\end{displaymath}
Write $\Lambda^e[T] = \cup_{i \neq e}\partial^{i \neq e} \Omega[T]$.

By the definition of dendroidal nerve, this is equivalently a diagram

\begin{displaymath}
\itexarray{
    \cup_{i \neq e } W(\partial^i \Omega[T]) &\to& X
    \\
    \downarrow & \nearrow_{\mathrlap{\hat \psi}}
    \\
    W(\Omega[T])
  }
  \,.
\end{displaymath}
The undetermined component to fill is that corresponding to the subtree $\tau$ of $T$ which is $T$ itself. According to prop. \ref{ComponentDescriptionofBVofTrees} on this the operad $W(\Omega[T])$ has the component

\begin{displaymath}
H^{\otimes i(\tau)} \simeq H^{\otimes i(\tau)- \{e\}}\otimes H
  \,.
\end{displaymath}
The map $\hat \psi$ has to send this into $X$ while being compatible with the given faces. By prop. \ref{BVResolutionAsFunctorOnOmega} this means that its precomposition with all the inclusions

\begin{displaymath}
(id, \cdots, id, i_0, id, \cdots, id) \otimes id
  : 
  H \otimes \cdots \otimes H \otimes I \otimes H \otimes \cdots \otimes H \otimes H
  \to 
  H^{\otimes i(\tau)- \{e\}}\otimes H
\end{displaymath}
is fixed. Moreover, the assignment needs to be compatible with the composition operations, which by prop. \ref{CompositioninWT} means that also the precomposition with all the maps

\begin{displaymath}
(id, \cdots, id, i_1, id, \cdots, id)
  : 
  H \otimes \cdots \otimes H \otimes I \otimes H \otimes \cdots \otimes H
  \to 
  H^{\otimes i(\tau)}
\end{displaymath}
is fixed. In total this means that the components of $\hat \psi$ need to form an extension of the form

\begin{displaymath}
\itexarray{
    (\partial H^{\otimes i(\tau)- \{e\}})
    \otimes H
    \cup
    H^{\otimes i(\tau) - \{e\}}\otimes I
    &\to& X(\tau)
    \\
    \downarrow & \nearrow
    \\
    H^{\otimes i(\tau)}
  }
\end{displaymath}
in $\mathcal{E}$, where

\begin{displaymath}
\partial H^n := 
    (I \coprod I) \otimes H^{n-1}
    \coprod
    H (I \coprod I) \otimes H^{n-2}
    \coprod
    \cdots
    \stackrel{(i_0,i_1)\otimes id \coprod \cdots}{\to}
    H^n
  \,.
\end{displaymath}
One sees that the left vertical morphism is an acyclic cofibration, by the [[pushout-product axiom]] in the [[monoidal model category]] $\mathcal{E}$. Therefore by the assumption that $X(\tau)$ is fibrant, such a lift does exist.

\end{proof}
\hypertarget{LeftAdjoint}{}\subsubsection*{{Left adjoint}}\label{LeftAdjoint}

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

\begin{displaymath}
W_! : dSet \to sSet Operad
\end{displaymath}
for the [[Yoneda extension]] of

\begin{displaymath}
\Omega \hookrightarrow Operad \hookrightarrow sSet Operad \stackrel{W}{\to} sSet Operad
  \,;
\end{displaymath}
hence for the [[functor]] from [[dendroidal sets]] to [[simplicial operads]], which

\begin{itemize}%
\item preserves [[colimits]];


\item on trees $\Omega \hookrightarrow Operad \hookrightarrow sSet Operad$ is given by the [[Boardman-Vogt resolution]] as discussed \hyperlink{BVResolutionOfTrees}{above}.



\end{itemize}
\end{defn}
By the general lore of [[nerve and realization]] we have

\begin{prop}
\label{}\hypertarget{}{}
$W_!$ is [[left adjoint]] to $hcN_d$

\begin{displaymath}
(W_! \dashv hcN_d)
  : 
  sSet Operad
  \stackrel{\overset{N_d}{\leftarrow}}{\underset{hcN_d}{\to}}
 dSet
  \,.
\end{displaymath}
\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
For $P \in sSet Operad$, the [[unit of an adjunction|counit]]

\begin{displaymath}
W_! hcN_d (P) \to P
\end{displaymath}
is essentially the [[Boardman-Vogt resolution]] of $P$.

For a cofibrant and fibrant $X \in dSet$, the [[unit of an adjunction|unit]]

\begin{displaymath}
X \to hcN_d W_!(X)
\end{displaymath}
may be viewed as a ``strictification'' of the [[(infinity,1)-operad]] given by $X$, in that $W_!(X)$, being a simplicial operad, has strictly associative composition.

\end{remark}
\hypertarget{relation_to_ordinary_dendroidal_nerve}{}\subsubsection*{{Relation to ordinary dendroidal nerve}}\label{relation_to_ordinary_dendroidal_nerve}

By the general properties of the [[Boardman-Vogt resolution]] (but also immediately checked directly) we have

\begin{prop}
\label{CounitOfResolutionOnTrees}\hypertarget{CounitOfResolutionOnTrees}{}
There is a [[natural transformation]]

\begin{displaymath}
\epsilon : W \Rightarrow \Omega(-) : \Omega \to sSet Operad
\end{displaymath}
\begin{displaymath}
\epsilon_T : W(T) \to T
\end{displaymath}
(natural in the tree $T \in \Omega$), which is a bijection on colors and is on the components of prop. \ref{ComponentDescriptionofBVofTrees} the canonical map

\begin{displaymath}
\Delta[1]^{i(V)} \to *
  \,.
\end{displaymath}
Each $\epsilon_T$ is hence a weak equivalence of simplicial operads. In particular

\begin{displaymath}
\pi_0(W(T)) \to T
\end{displaymath}
is an [[isomorphism]].

\end{prop}
This induces hence a [[natural transformation]]

\begin{displaymath}
W_! \Rightarrow \tau_d : dSet \to sSet Operad
\end{displaymath}
to the left adjoint $\tau_d$ of the ordinary [[dendroidal nerve]] (the ``fundamental operad'' construction).

\begin{prop}
\label{}\hypertarget{}{}
For every [[dendroidal set]] $X$, the natural morphism

\begin{displaymath}
\pi_0 W_!(X) \to \pi_0 \tau_d (X) = \tau_d(X)
\end{displaymath}
is an [[isomorphism]] of simplicial operads.

\end{prop}
This appears as \hyperlink{CisinskiMoerdijk}{Cisinski-Moerdijk, prop. 4.4}.

\begin{proof}
The functors $\pi_0 W_!$ and $\tau_d$, being [[left adjoints]], both preserve small [[colimits]]. Therefore it is sufficient to check the statement for $X = \Omega[T]$ a tree. There it is prop. \ref{CounitOfResolutionOnTrees}.

\end{proof}
\hypertarget{quillen_equivalence}{}\subsubsection*{{Quillen equivalence}}\label{quillen_equivalence}

\begin{theorem}
\label{}\hypertarget{}{}
The [[adjunction]] $(W_! \dashv hcN_d)$ from \hyperlink{LeftAdjoint}{above} is a [[Quillen equivalence]] between the [[model structure on operads]] over [[Top]]/[[sSet]] and the [[model structure on dendroidal sets]].

\end{theorem}
We discuss some input to this statement.

\begin{prop}
\label{WIsLeftQuillen}\hypertarget{WIsLeftQuillen}{}
The functor $W_! : dSet \to sSet Operad$ sends normal monomorphisms to cofibrations, and inner [[anodyne extensions]] to [[acyclic cofibrations]] in the [[model structure on sSet-operads]].

\end{prop}
This appears as \hyperlink{CisinskiMoerdijk}{Cisinski-Moerdijk, prop. 4.5}.

\begin{proof}
Observe that the morphism classes in question are, as discussed at \emph{[[dendroidal set]]}, the saturated classes generated by the dendroidal boundary inclusions and by the dendroidal horn inclusions, respectively.

Since $W_!$ is left adjoint, it therefore suffices to check the statement on these generating inclusions. Moreover, by construction, on trees $W_!$ coincides with the [[Boardman-Vogt resolution]] of the operads free on these trees.

It follows that the generating inclusions are sent by $W_!$ to morphisms of simplicial operads which are

\begin{itemize}%
\item bijective on objects;


\item isomorphisms on all but one simplicial set of operations: that corresponding to the maximal subtree;


\item on this remaining simplicial set of operations a product of identities with cofibrations of simplicial sets (monomorphisms), and following through the combinatorics shows that these are acyclic for the case of anodyne extensions.



\end{itemize}
It follows that these morphisms of simplicial operads have the left lifting property again operation-object-wise [[Kan fibrations]] (there is no further composition to be respected, since the maximal subtree operation has no further non-trivial composites), and hence against the fibrations of the [[model structure on sSet-operads]].

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
Prop.\ref{hcNOfLocllyFibrantOperadIsFibrant} is, in turn, a direct consequence of this.

\end{remark}
\hypertarget{resolution_and_rectification}{}\subsubsection*{{Resolution and rectification}}\label{resolution_and_rectification}

\begin{prop}
\label{}\hypertarget{}{}
Let $P$ be an [[operad]] in [[Set]], regarded as an $\mathcal{E}$-operad. Then the $(W_! \dashv hcN_d)$-[[unit of an adjunction|counit]]

\begin{displaymath}
W_! N_d(P) = W_! hcN_d (P) \to P
\end{displaymath}
is isomorphic to the [[Boardman-Vogt resolution]] $W_H(P)$ of $P$.

In particular, therefore, there is a [[natural isomorphism]]

\begin{displaymath}
Hom_{\mathcal{E}Operad}(W_H(P), Q)
  \simeq
  Hom_{dSet}(N_d(P), hcN_d(Q))
  \,.
\end{displaymath}
\end{prop}
(Here we are using that on a discrete operad $P$ the homotopy coherent dendroidal nerve trivially coincides with the ordinary dendroidal nerve $N_d$.)

\begin{proof}
By inspection of the relevant formulas.

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
For a cofibrant and fibrant dendroidal set $X$, the $(W_! \dashv hcN_d)$-[[unit of an adjunction|unit]]

\begin{displaymath}
X \to hcN_d W_! X
\end{displaymath}
is an equivalence.

\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
Since composition of operations in a simplicial operad is strictly associative, this may be understood as producing a semi-strictification of the $\infty$-operad $X$.

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

[[!include table - models for (infinity,1)-operads]]

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

The fact that the homotopy coherent nerve if a locally fibrant operad is inner Kan is shown in section 7 of

\begin{itemize}%
\item [[Ieke Moerdijk]], [[Ittay Weiss]], \emph{On inner Kan complexes in the category of dendroidal sets}, (\href{http://arxiv.org/abs/math/0701295}{math.CT/0701295})

\end{itemize}
The Quillen adjunction properties of the homotopy coherent dendroidal nerve are discussed in section 4 of

\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}
Lecture notes on these two topics are in section 6 and 9 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.cat/HigherCategories/hc1.pdf}{preliminary writeup})

\end{itemize}
[[!redirects homotopy coherent dendroidal nerve]]



\end{document}