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\begin{document}

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\section*{(infinity,1)-operad}

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

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

[[!include higher algebra - contents]]

\hypertarget{category_theory}{}\paragraph*{{$(\infty,1)$-Category theory}}\label{category_theory}

[[!include quasi-category theory contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{in_terms_of_dendroidal_sets}{In terms of dendroidal sets}\dotfill \pageref*{in_terms_of_dendroidal_sets} \linebreak
\noindent\hyperlink{InTermsOfInfinityCategoriesOfOperators}{In terms of $(\infty,1)$-categories of operators}\dotfill \pageref*{InTermsOfInfinityCategoriesOfOperators} \linebreak
\noindent\hyperlink{BasicDefinitions}{Basic definitions}\dotfill \pageref*{BasicDefinitions} \linebreak
\noindent\hyperlink{ModelForinfOpera}{Model for $(\infty,1)$-categories of operators}\dotfill \pageref*{ModelForinfOpera} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation}{Relation between the two definitions}\dotfill \pageref*{relation} \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 notion of \emph{$(\infty,1)$-operad} is to that of [[(∞,1)-category]] as [[operad]] is to [[category]].

So, roughly, an $(\infty,1)$-operad is an algebraic structure that has for each given type of input and one type of output an [[∞-groupoid]] of operations that take these inputs to that output.

There is a fairly evident notion of [[∞-algebra over an (∞,1)-operad]]s. Examples include

\begin{itemize}%
\item [[E-∞ algebra]]s


\item [[L-∞ algebra]]s;


\item [[A-∞ algebra]]s.



\end{itemize}
$(\infty,1)$-Operads form an [[(∞,2)-category]] [[(∞,1)Operad]].

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

Two models for $(\infty,1)$-operads exist to date, one by [[Denis-Charles Cisinski|Cisinski]]--[[Ieke Moerdijk|Moerdijk]]--[[Ittay Weiss|Weiss]], the other by [[Jacob Lurie|Lurie]]. It is expected though not yet entirely proved that the two are equivalent ([[Higher Algebra]], draft, Remark 2.0.0.8).

The first one models $(\infty,1)$-operads as [[dendroidal set]]s in close analogy to (in fact as a generalization of how) [[simplicial set]]s model [[(∞,1)-category|(∞,1)-categories]].

The second models the [[(∞,1)-category]] version of a [[category of operators]] of an operad.

\hypertarget{in_terms_of_dendroidal_sets}{}\subsubsection*{{In terms of dendroidal sets}}\label{in_terms_of_dendroidal_sets}

Here [[simplicial set]]s are generalized to [[dendroidal set]]s. The theory of $(\infty,1)$-operads is then formulated in terms of dendroidal sets in close analogy to how the theory of [[(∞,1)-category|(∞,1)-categories]] is formulated in terms of [[simplicial set]]s.

There is a [[model structure on dendroidal sets]] whose fibrant objest are the \textbf{quasi-operad}s in direct analogy to the notion of [[quasi-category]].

So the [[model structure on dendroidal sets]] is a [[presentable (∞,1)-category|presention]] of the [[(∞,1)-category]] of $(\infty,1)$-operads. It is [[Quillen equivalence|Quillen equivalent]] to the standard [[model structure on operads]] enriched over [[Top]] or [[sSet]]. Therefore, conversely, the traditional homotopy-theoretic constructions on topological and chain operads (such as cofibrant [[resolution]]s in order to present homtopy algebras such as [[A-∞ algebra]]s, [[L-∞ algebra]]s, [[homotopy BV-algebra]]s and the like) are also indeed presentations of $(\infty,1)$-operads.

\hypertarget{InTermsOfInfinityCategoriesOfOperators}{}\subsubsection*{{In terms of $(\infty,1)$-categories of operators}}\label{InTermsOfInfinityCategoriesOfOperators}

Every [[operad]] $A$ encodes and is encoded by its [[category of operators]] $C_A$. In the approach to $(\infty,1)$-operators described below, the notion of category of operators is generalized to an [[(∞,1)-category]] of operators.

In this approach an $(\infty,1)$-operad $C^\otimes$ is regarded as an [[(∞,1)-category]] $C$ -- the unary part of the $(\infty,1)$-operad to be described-- with extra structure that determines [[(∞,1)-functor]]s $C^{\times n} \to C$.

This and the conditions on these are encoded in requiring that $C^\otimes$ is an $(\infty,1)$-functor $C^\otimes \to \Gamma$ over [[Segal Gamma-category|Segal's category]] $\Gamma$ of pointed finite sets, satisfying some conditions.

In particular, any [[symmetric monoidal (∞,1)-category]] yields an example of an $(\infty,1)$-operad in this sense. In fact, symmetric monoidal $(\infty,1)$-categories can be \emph{defined} as $(\infty,1)$-operads such that the functor $C^\otimes \to \Gamma$ is a [[coCartesian fibration]]. (For the moment, see [[monoidal (infinity,1)-category]] for more comments and references on higher operads in this context.)

This is the approach described in (\hyperlink{LurieCommutative}{LurieCommutative})

\hypertarget{BasicDefinitions}{}\paragraph*{{Basic definitions}}\label{BasicDefinitions}

We are to generalize the following construction from [[categories]] to [[(∞,1)-categories]].

\begin{defn}
\label{}\hypertarget{}{}
For $\mathcal{O}$ a [[symmetric multicategory]], write $\mathcal{O}^\otimes \to FinSet^{*/}$ for its [[category of operators]].

Here $\mathcal{O}^\otimes$ is the [[category]] whose

\begin{itemize}%
\item [[objects]] are finite sequences ([[tuples]]) of objects of $\mathcal{O}$;


\item [[morphisms]] $(X_1, \cdots, X_{n_1}) \to (Y_1, \cdots, Y_{n_2})$ are given by a morphism $\alpha \colon \langle n_1\rangle \to \langle n_2\rangle$ in $FinSet_*$ together with a collection of [[multimorphisms]]

\begin{displaymath}
\left\{
    \phi_j \in \mathcal{O}\left( \left\{ X_i\right\}_{i \in \alpha^{-1}\left\{j\right\}} , Y_j \right)
  \right\}_{1 \leq j \leq n_2}
  \,.
\end{displaymath}


\end{itemize}
The [[functor]] $p \colon \mathcal{O}^\otimes \to FinSet^{*/}$ is the evident [[forgetful functor]].

\end{defn}
In (\hyperlink{Lurie}{Lurie}) this is construction 2.1.1.7.

This motivates the following definition of the generalization of this situation to [[(∞,1)-category theory]].

\begin{defn}
\label{FinSetPointed}\hypertarget{FinSetPointed}{}
Write $FinSet^{*/}$ for the category of [[pointed object|pointed]] [[finite set]] ([[Segal Gamma-category|Segal's Gamma-category]]).

For $n \in \mathbb{N}$ we write

\begin{displaymath}
\langle n\rangle
  \coloneqq
  {*} \coprod [n]
  \in 
  FinSet^{*/}
\end{displaymath}
for the [[pointed set]] with $n+1$ elements.

A morphism in $FinSet^{*/}$

\begin{itemize}%
\item is called an \textbf{[[inert morphism]]} if it is a surjection, and an [[injection]] on those elements that are not sent to the base point. That is, the preimage of every non-base point is a singleton.


\item called an \textbf{[[active morphism]]} if only the basepoint goes to the basepoint.



\end{itemize}
For $n \in \mathbb{N}$ and $1 \leq i \leq n$ write

\begin{displaymath}
\rho^i \colon \langle n\rangle \to \langle 1\rangle
\end{displaymath}
for the inert morphism that sends all but the $i$th element to the basepoint.

Notice that for each $n \in \mathbb{N}$ there is a unique [[active morphism]] $\langle n\rangle \to \langle 1\rangle$.

\end{defn}
(\hyperlink{Lurie}{Lurie, def. 2.1.1.8})

\begin{defn}
\label{InfinityOperadByCategoryOfOperators}\hypertarget{InfinityOperadByCategoryOfOperators}{}
The \textbf{$(\infty,1)$-category of operators of an $(\infty,1)$-operad} is a morphism

\begin{displaymath}
p \colon \mathcal{O}^\otimes \to FinSet^{*/}
\end{displaymath}
of [[quasi-categories]] such that the following conditions hold:

\begin{enumerate}%
\item For every [[inert morphism]] in $FinSet^{*/}$ and every [[object]] over it, there is a lift to a $p$-[[coCartesian morphism]] in $\mathcal{O}^\otimes$. In particular, for $f \colon \langle n_1\rangle \to \langle n_2\rangle$ inert, there is an induced [[(∞,1)-functor]]

\begin{displaymath}
f_! 
    \colon 
  \mathcal{O}^\otimes_{\langle n_1\rangle}
   \to
  \mathcal{O}^\otimes_{\langle n_2\rangle}
  \,.
\end{displaymath}

\item The coCartesian lifts of the inert projection morphisms induce an equivalence of [[derived hom-spaces]] in $\mathcal{O}^{\otimes}$ between maps into multiple objects and the products of the maps into the separete objects:

For $f \colon \langle n_1 \rangle \to \langle n_2 \rangle$ write $\mathcal{O}^\otimes_f(-,-) \hookrightarrow \mathcal{O}^\otimes(-,-)$ for the components of the [[derived hom-space]] covering $f$, then the $(\infty,1)$-functor

\begin{displaymath}
\mathcal{O}^\otimes_f(C_1,C_2)
  \to
  \underset{1 \leq k \leq n_2}{\prod}
  \mathcal{O}^\otimes_{\rho^i\circ f}(C_1,(C_2)_i)
\end{displaymath}
induced as above is an [[equivalence of infinity-groupoids|equivalence]].


\item For every finite collection of objects $C_1, \cdots c_n \in \mathcal{O}^\otimes_{\langle 1\rangle}$ there exists a multiobject $C \in \mathcal{O}^\otimes_{\langle n\rangle}$ and a collection of $p$-[[coCartesian morphisms]] $\{C \to C_i\}$ covering $\rho^i$.

Equivalently (given the first two conditions): for all $n \in \mathbb{N}$ the $(\infty,1)$-functors $\{(\rho^i)_!\}_{1 \leq i \leq n}$ induce an [[equivalence of (∞,1)-categories]]

\begin{displaymath}
\mathcal{O}^\otimes_{\langle n\rangle}
  \to
  (\mathcal{O}^\otimes_{\langle 1\rangle})^{\times^n}
\end{displaymath}


\end{enumerate}
\end{defn}
(\hyperlink{Lurie}{Lurie, def. 2.1.1.10, remark 2.1.1.14})

\begin{remark}
\label{}\hypertarget{}{}
The conditions in def. \ref{InfinityOperadByCategoryOfOperators} mean that $p \colon \mathcal{O}^\otimes \to FinSet^{*/}$ encodes

\begin{enumerate}%
\item an [[(∞,1)-category]] $\mathcal{O} \coloneqq \mathcal{O}^\otimes_{\langle 1\rangle}$;


\item for each $n \in \mathbb{N}$ an $n$-ary operation given by the $(\infty,1)$-functor

\begin{displaymath}
\mathcal{O}^{n}
  =
  (\mathcal{O}^\otimes_{\langle 1\rangle})^{\times n}
  \simeq
  \mathcal{O}^{\otimes}_{\langle n\rangle}
  \to 
  \mathcal{O}^\otimes_{\langle 1\rangle}
  = 
  \mathcal{O}
\end{displaymath}
induced by the unique [[active morphism]] $\langle n\rangle \to \langle 1\rangle$


\item a coherently associative multicomposition of these operations.



\end{enumerate}
\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The notion of def. \ref{InfinityOperadByCategoryOfOperators} may also be called a \textbf{symmetric $(\infty,1)$-multicategory} or \textbf{colored $(\infty,1)$-operad}. The \emph{colors} are the [[objects]] of $\mathcal{O}$.

\end{remark}
We now turn to the definition of [[homomorphisms]] of $(\infty,1)$-operads.

\begin{defn}
\label{InertMorphismsInInfinityOperad}\hypertarget{InertMorphismsInInfinityOperad}{}
Given an $(\infty,1)$-operad $p \colon \mathcal{O}^\otimes \to FinSet^{*/}$ as in def. \ref{InfinityOperadByCategoryOfOperators}, a [[morphism]] $f$ in $\mathcal{O}^\otimes$ is called an \textbf{inert morphism} if

\begin{enumerate}%
\item $p(f)$ is an [[inert morphism]] in $FinSet^{*/}$ by def. \ref{FinSetPointed};


\item $f$ is a $p$-[[coCartesian morphism]].



\end{enumerate}
\end{defn}
\begin{defn}
\label{MorphismOfInfinityOperads}\hypertarget{MorphismOfInfinityOperads}{}
A \textbf{[[morphism of (∞,1)-operads]]} is a map between their [[(∞,1)-categories of operators]] over $FinSet^{*/}$ that preserves the inert morphisms of def. \ref{InertMorphismsInInfinityOperad}.

\end{defn}
Morphisms of operads $\mathcal{O}_1 \to \to \mathcal{O}_2$ can be understood equivalently as exhibiting an $\mathcal{O}_1$-[[algebra over an operad|algebra]] in $\mathcal{O}_2$. Therefore:

\begin{defn}
\label{}\hypertarget{}{}
For $\mathcal{O}_1, \mathcal{O}_2$ to $(\infty,1)$-operads, write

\begin{displaymath}
Alg_{\mathcal{O}_1}(\mathcal{O}_2)
  \hookrightarrow
  qCat_{/FinSet^{*/}}(\mathcal{O}_1, \mathcal{O}_2)
\end{displaymath}
for the full [[sub-(∞,1)-category]] of the [[(∞,1)-functor (∞,1)-category]] on those that are [[morphisms of (∞,1)-operads]] by def. \ref{MorphismOfInfinityOperads}.

\end{defn}
(\hyperlink{Lurie}{Lurie, def. 2.1.2.7}).

We also have the notion of

\begin{itemize}%
\item \emph{[[fibration of (∞,1)-operads]]};


\item \emph{[[coCartesian fibration of (∞,1)-operads]]}.



\end{itemize}
See there for more details.

\hypertarget{ModelForinfOpera}{}\paragraph*{{Model for $(\infty,1)$-categories of operators}}\label{ModelForinfOpera}

There is a [[model category]] that [[presentable (infinity,1)-category|presents]] the [[(∞,1)-category]] $(\infty,1)Cat_{Oper}$ of $(\infty,1)$-categories of operations.

\begin{prop}
\label{}\hypertarget{}{}
There exists a

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


\item [[combinatorial model category|combinatorial]]



\end{itemize}
[[model category]] $\mathcal{P} Op_{(\infty,1)}$

\begin{itemize}%
\item whose underlying [[category]] has

\begin{itemize}%
\item [[object]]s are [[marked simplicial set]] $S$ equipped with a morphism $S \to N(FinSet_*)$ such that marked edges map to inert morphisms in $FinSet_*$ (those for which the preimage of the marked point contains just the marked point)


\item [[morphism]]s are morphisms of [[marked simplicial set]]s $S \to T$ such that the triangle

\begin{displaymath}
\itexarray{
    S &&\to&& T
    \\
    & \searrow && \swarrow
    \\
    && N(FinSet_*)
  }
\end{displaymath}
commutes;



\end{itemize}

\item which is canonically an [[SSet]]-[[enriched category]];


\item and whose [[model category|model structure]] is given by

\begin{itemize}%
\item cofibrations are those morphisms whose underlying morphisms of [[simplicial set]]s ate cofibrations, hence [[monomorphism]]s


\item weak equivalences are those morphisms $S \to T$ such that for all $A \to N(FinSet_*)$ that are $(\infty,1)$-categories of operations by the above definition, the morphism of [[SSet]]-[[hom object]]s

\begin{displaymath}
\mathcal{P}Op_\infty(T,A) \to 
  \mathcal{P}Op_\infty(S,A)
\end{displaymath}
is a homotopy equivalence of simplicial sets.


\item an object is fibrant if and only if it is an $(\infty,1)$-category of operations, by the above definition.



\end{itemize}


\end{itemize}
\end{prop}
This is prop 1.8 4 in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{Commutative algebra} (\href{http://www.math.harvard.edu/~lurie/papers/DAG-III.pdf}{pdf})

\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

We list some examples of $(\infty,1)$-operads incarnated as their [[(∞,1)-categories of operators]] by def. \ref{InfinityOperadByCategoryOfOperators}.

The first basic examples to follow are in fact all given by [[1-categories]] [[category of operators|of operators]].

\begin{example}
\label{}\hypertarget{}{}
The [[identity]] functor on $FinSet^{*/}$ exhibits an $(\infty.1)$-operad. This is the \textbf{[[commutative operad]]}

\begin{displaymath}
Comm^\otimes = FinSet^{*/} \stackrel{id}{\to} FinSet^{*/}
  \,.
\end{displaymath}
The [[(∞,1)-algebras over an (∞,1)-operad]] over this $(\infty,1)$-operad are [[E-∞ algebras]].

\end{example}
\begin{example}
\label{}\hypertarget{}{}
The \textbf{[[associative operad]]} has $Assoc^\otimes$ the category whose objects are the natural numbers, whose $n$-ary operations are labeled by the [[total orders]] on $n$ elements, equivalently the elements of the [[symmetric group]] $\Sigma_n$, and whose composition is given by forming consecutive total orders in the obvious way.

The [[(∞,1)-algebras over an (∞,1)-operad]] over this $(\infty,1)$-operad are [[A-∞ algebras]]

\end{example}
In (\hyperlink{Lurie}{Lurie}) this is remark 4.1.1.4.

\begin{example}
\label{}\hypertarget{}{}
The \textbf{[[operad for modules over an algebra]]} $LM$ is the [[colored operad|colored]] [[symmetric operad]] whose

\begin{itemize}%
\item [[objects]] are two elements, to be denoted $\mathfrak{a}$ and $\mathfrak{n}$;


\item [[multimorphisms]] $(X_i)_{i = 1}^n \to Y$ form

\begin{itemize}%
\item if $Y = \mathfrak{a}$ and $X_i = \mathfrak{a}$ for all $i$ then: the set of [[linear orders]] on $n$ elements, equivalently the elements of the [[symmetric group]] $\Sigma_n$;


\item if $Y = \mathfrak{n}$ and exactly one of the $X_i = \mathfrak{n}$ then: the set of linear order $\{i_1 \lt \cdots \lt i_n\}$ such that $X_{i_n} = \mathfrak{n}$


\item otherwise: the empty set;



\end{itemize}

\item [[composition]] is given by composition of linear orders as for the [[associative operad]].



\end{itemize}
The [[(∞,1)-algebras over an (∞,1)-operad]] over this $(\infty,1)$-operad are pairs consisting of [[A-∞ algebras]] with [[(∞,1)-modules]] over them.

\end{example}
In (\hyperlink{Lurie}{Lurie}) this appears as def. 4.2.1.1.

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{[[operad for bimodules over algebras]]} $BMod$ is the [[colored operad|colored]] [[symmetric operad]] whose

\begin{itemize}%
\item [[objects]] are three elements, to be denoted $\mathfrak{a}_-, \mathfrak{a}_+$ and $\mathfrak{n}$;


\item [[multimorphisms]] $(X_i)_{i = 1}^n \to Y$ form

\begin{itemize}%
\item if $Y = \mathfrak{a}_-$ and all $X_i = \mathfrak{a}_-$ then: the set of [[linear orders]] of $n$ elements;


\item if $Y = \mathfrak{a}_*$ and all $X_i = \mathfrak{a}_*$ then again: the set of [[linear orders]] of $n$ elements;


\item if $Y = \mathfrak{n}$: the set of linear orders $\{i_1 \lt \cdots \lt i_n\}$ such that there is exactly one index $i_k$ with $X_{i_k} = \mathfrak{n}$ and $X_{i_j} = \mathfrak{a}_-$ for all $j \lt k$ and $X_{i_j} = \mathfrak{a}_+$ for all $k \gt k$.



\end{itemize}

\item [[composition]] is given by the composition of linear orders as for the [[associative operad]].



\end{itemize}
The [[(∞,1)-algebras over an (∞,1)-operad]] over this $(\infty,1)$-operad are pairs consisting of two [[A-∞ algebras]] with an [[(∞,1)-bimodule]] over them.

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

\hypertarget{relation}{}\subsubsection*{{Relation between the two definitions}}\label{relation}

At the time of this writing there is no discussion in ``the literature'' of the relation between the definition of $(\infty,1)$-operads in terms of dendroidal sets (Cisinski, Moerdijk, Weiss) and $(\infty,1)$-categories of operators (Lurie). The following are some tentative observations. - [[Urs Schreiber|Urs]]

update: meanwhile this has been worked out by some people. Results should appear in preprint form soon.

There is an obvious way to regard a [[tree]] as an $(\infty,1)$-category of operators:

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(dendroidal $(\infty,1)$-category of operators)}

Let

\begin{displaymath}
\omega : \Omega \hookrightarrow Op \stackrel{C_{(-)}}{\to}
  Cat/FinSet_* \stackrel{N}{\to}
  \mathcal{P}Op_{(\infty,1)}
\end{displaymath}
be the dendroidal object given by the following composition:

\begin{itemize}%
\item $\Omega \hookrightarrow Op$ is the functor from the [[tree category]] $\Omega$ to the category of symmetric colored [[operad]]s (over [[Set]]) that sends a tree to the operad freely generated from it;


\item $Op \stackrel{C_{(-)}}{\to} Cat/FinSet_*$ sends an [[operad]] to its [[category of operators]];


\item $Cat/FinSet_* 
\stackrel{N}{\to} \mathcal{P}Op_{(\infty,1)}$ takes the [[nerve]] of this category, regarded as a [[marked simplicial set]] over $N(FinSet_*)$, whose marked edges are the inert morphisms in the category of operations.



\end{itemize}
\end{defn}
Following the general pattern of [[nerve and realization]], we get:

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(dendroidal nerve of Lurie-$\infty$-operad)}

The functor

\begin{displaymath}
N_d := 
   Hom_{\mathcal{P}Op_{(\infty,1)}}(\omega(-), -):
   \mathcal{P}Op_{(\infty,1)} \to dSet
\end{displaymath}
that sends a [[marked simplicial set]] $A \to N(FinSet_*)$ to the [[dendroidal set]] which sends a [[tree]] $T$ to the set of morphisms of $\omega(T)$ into $A$

\begin{displaymath}
N_d(A) : T \mapsto Hom_{\mathcal{P}Op_{(\infty,1)}}(\omega(T), A)
\end{displaymath}
is the \textbf{dendroidal nerve} of $A$.

\end{defn}
One expects that $N_d$ induces a [[Quillen adjunction]] and indeed a [[Quillen equivalence]] between the above model category structure on $\mathcal{P}Op_{(\infty,1)}$ and the [[model structure on dendroidal sets]]. The following is as far as I think I can prove aspects of this. -[[Urs Schreiber|Urs]].

\begin{prop}
\label{}\hypertarget{}{}
The dendroidal nerve functor has the following properties:

\begin{itemize}%
\item it is the [[right adjoint]] of a [[SSet]]-[[enriched functor|enriched]] [[adjunction]]

\begin{displaymath}
C_{(-)} : dSet \stackrel{\leftarrow}{\to} \mathcal{P}Op_{(\infty,1)}
 : N_d
\end{displaymath}

\item it sends fibrant objects to fibrant objects

i.e. it sends $(\infty,1)$-categories of operations to $(\infty,1)$-operads in their incarnation as ``quasi-operads'';


\item it sends objects $\pi : A \to N(FinSet_*)$ that come from grouplike [[symmetric monoidal (infinity,1)-category|symmetric monoidal]] [[∞-groupoid]]s to fully Kan dendroidal sets (that have the extension property with respect to all horns)


\item it sends objects $\pi : A \to N(FinSet_*)$ that come from [[symmetric monoidal (infinity,1)-category|symmetric monoidal (∞,1)-categories]] to dendroidal sets that have the extension property with respect to at least one outer horn $\Lambda_{v} T$ for $v \in T$ an $n$-corolla, for all $n \in \mathbb{N}$.


\item its [[left adjoint]] sends cofibrations to cofibrations and acyclic cofibrations with cofibrant domain to acyclic cofibrations.



\end{itemize}
\end{prop}
\begin{proof}
\textbf{respect for fibrant objects}. If $A \to N(FinSet_*)$ is fibrant, then in particual $A$ is a [[weak Kan complex]] hence has the extension property with respect to all inner [[horn]] inclusions of [[simplex|simplices]]. We need to show that this implies that $N_d(A)$ has the extension property with respect to all inner horn inclusions of [[tree]]s.

By an (at the moment unpublished) result by [[Ieke Moerdijk|Moerdijk]], right [[lifting property]] with respect to inner horn inclusions of trees is equivalend to right lifting property with respect to inclusions of [[spine]]s of trees: the union over all the corollas in a tree.

For this the extension property means that if we find a collection $\{C_{k_i} \to N_d(A)\} = Sp(T)$ of corollas in $N_d(A)$ that match at some inputs and output, then these can be composed to an image $T \to N_d(A)$ of the corresponding tree $T$ in $N_d(A)$.

An image of $T$ in $N_d(A)$ is an image of $\omega(T)$ in $A$. In the [[category of operators]] $\omega(A)$ every tree may be represented as the composite of a sequence of morphisms each of which consists of precisely one of the corollas $C_{k_i}$ in parallel to identity morphisms. This way gluing the tree from the corollas is a matter of composing a sequence of edges in $A$. But this is guaranteed to be possible if $A$ is a [[weak Kan complex]].

\textbf{symmetric monoidal product and outer horn lifting}

As described at [[cartesian morphism]], an edge $f : \Delta^1 \to A$ in $A$ is coCartesian if for all diagrams

\begin{displaymath}
\itexarray{
    \Delta^{0,1}
    \\
    \downarrow & \searrow^f
    \\
    \Lambda^n_0 &\to & A
    \\
    \downarrow && \downarrow
    \\
    \Delta^n &\to& N(FinSet_*)
  }
\end{displaymath}
of 0-horn lifting problems where the first edge of the horn is $f$ itself, there exists a lift

\begin{displaymath}
\itexarray{
    \Delta^{0,1}
    \\
    \downarrow & \searrow^f
    \\
    \Lambda^n_0 &\to & A
    \\
    \downarrow &\nearrow & \downarrow
    \\
    \Delta^n &\to& N(FinSet_*)
  }
  \,.
\end{displaymath}
For $f$ the parallel application of an $n$-corolla with a collection of identity morphisms this implies that any outer horn $\Lambda_v T \to N_d(A)$ for which the vertex $v : C_n \to N_d(A)$ maps to $f$, the dendroidal set $N_d(A)$ has the extension property with respect to the inclusion $\Lambda_d T \hookrightarrow T$.

\textbf{the left adjoint and its respect for cofibrations}

By general nonsense the [[left adjoint]] to $N_d$ is given by the [[coend]]

\begin{displaymath}
C_{(-)} : dSet \to \mathcal{P}Op_{(\infty,1)}
\end{displaymath}
\begin{displaymath}
C_P = \int^{T \in \Omega} \omega(T) \cdot P(T)
  \,,
\end{displaymath}
where in the integrand we have the tautological [[copower|tensoring]] of $\mathcal{P}Op_{(\infty,1)}$ over [[Set]].

Notice that $\omega : \Omega \to \mathcal{P}Op_{(\infty,1)}$ is an [[SSet]]-[[enriched functor]] for the ordinary category $\Omega$ regarded as a simplicially enriched category by the canonical embedding $Set \hookrightarrow SSet$. Therefore this adjunction $F \dashv N_d$ is defined entirely in [[SSet]]-[[enriched category theory]] and hence is a simplicial adjunction.

The [[model structure on dendroidal sets]] has a set of [[cofibrantly generated model category|generating cofibrations]] given by the boundary inclusions of trees. $\partial \Omega[T] \hookrightarrow \Omega[T]$. Tese evidenly map to monomorphisms of underlying simplicial sets under $F$, hence to cofibrations.

For $f : P \hookrightarrow Q$ an acyclic cofibration with cofibrant domain, we need to check that $C_f : C_X \to C_Y$ is a weak equivalence in $\mathcal{P}Op_{(\infty,1)}$. This is by definition the case if for every fibrant object $A$ the morphism

\begin{displaymath}
\mathcal{P}Op_{(\infty,1)}(C_Y,A) \to 
  \mathcal{P}Op_{(\infty,1)}(C_X,A)
\end{displaymath}
is a weak equivalence in the standard [[model structure on simplicial sets]]. By the simplicial adjunction $F \dashv N_d$ this is equivalent to

\begin{displaymath}
dSet(f,N_d(A))
  :
  dSet(Y,N_d(A)) \to 
  dSet(X,N_d(A))
\end{displaymath}
being a weak equivalence. By the above $N_d(A)$ is fibrant. By section 8.4 of the lecture notes on dendroidal sets cited at [[model structure on dendroidal sets]] a morphism between cofibrant dendroidal sets is a weak equivalence precisely if homming it into any fibrant dendroidal set produces an equivalence of homotopy categories.

Since $f$ is a weak equivalence between cofibrant objects by assumption, it follows that indeed $dSet(f,N_d(A))$ is a weak equivalence for all fibrant $A$.

\begin{quote}%
(AHM, or does it? there is a prob here, but I need to run now\ldots{})


\end{quote}
Hence $C_f$ is a weak equivalence.

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

\begin{itemize}%
\item [[table - models for (∞,1)-operads]]


\item [[algebraic theory]] / [[Lawvere theory]] / [[(∞,1)-algebraic theory]]


\item [[monad]] / [[(∞,1)-monad]]


\item [[operad]] / \textbf{$(\infty,1)$-operad}, [[model structure on operads]]

\begin{itemize}%
\item [[morphism of (∞,1)-operads]], [[fibration of (∞,1)-operads]]


\item [[generalized (∞,1)-operad]], [[family of (∞,1)-operads]]


\item [[algebra over an (∞,1)-operad]], [[model structure on algebras over an operad]]

\begin{itemize}%
\item [[module over an algebra over an (∞,1)-operad]], [[model structure on modules over an algebra over an operad]]

\end{itemize}

\item [[coherent (∞,1)-operad]]



\end{itemize}

\item [[operadic (∞,1)-Grothendieck construction]]


\item [[cohomology of operads]]



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

The formulation in terms of [[dendroidal sets]] is due to

\begin{itemize}%
\item [[Ieke Moerdijk]] [[Ittay Weiss]], \emph{Dendroidal sets} (\href{http://cat.inist.fr/?aModele=afficheN&cpsidt=20087314}{web})


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



\end{itemize}
Here are two blog entries on talks on this stuff:

\begin{itemize}%
\item \href{http://golem.ph.utexas.edu/category/2009/02/dendroidal_sets.html}{Dendroidal Sets and Infinity-Operads}


\item \href{http://golem.ph.utexas.edu/category/2009/02/moerdijk_on_infinityoperads.html}{Moerdijk on Infinity Operads}



\end{itemize}
The formulation in terms of an $(\infty,1)$-version of the [[category of operators]] is introduced in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Commutative Algebra]]} (\href{http://www.math.harvard.edu/~lurie/papers/DAG-III.pdf}{pdf})

\end{itemize}
and further discussed in

\begin{itemize}%
\item \emph{[[Ek-Algebras]]} .

\end{itemize}
Now in section 2 of the textbook

\begin{itemize}%
\item \emph{[[Higher Algebra]]}

\end{itemize}
The equivalence between the [[dendroidal set]]-formulation and the one in terms of $(\infty,1)$-categories of operators is shown in

\begin{itemize}%
\item [[Gijs Heuts]], [[Vladimir Hinich]], [[Ieke Moerdijk]], \emph{The equivalence between Lurie's model and the dendroidal model for infinity-operads} (\href{http://arxiv.org/abs/1305.3658}{arXiv:1305.3658})

\end{itemize}
Further equivalence to Barwick's complete Segal operads is discussed in

\begin{itemize}%
\item Hongyi Chu, [[Rune Haugseng]], [[Gijs Heuts]], \emph{Two models for the homotopy theory of $\infty$-operads} (\href{https://arxiv.org/abs/1606.03826}{arXiv:1606.03826})

\end{itemize}
For an account in terms of \emph{analytic} [[monads]], that is, monads that are cartesian (multiplication and unit transformations are cartesian) and the underlying endofunctor preserves sifted colimits and wide pullbacks (or equivalently all weakly contractible limits), see

\begin{itemize}%
\item [[David Gepner]], [[Rune Haugseng]], [[Joachim Kock]], \emph{∞-Operads as Analytic Monads}, (\href{https://arxiv.org/abs/1712.06469}{arXiv:1712.06469})

\end{itemize}
[[!redirects (∞,1)-operad]] [[!redirects (∞,1)-operads]] [[!redirects (infinity,1)-operads]]



\end{document}