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\begin{document}

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\section*{model structure on algebras over an operad}

\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{existence}{Existence}\dotfill \pageref*{existence} \linebreak
\noindent\hyperlink{cofibrant_operads}{Cofibrant operads}\dotfill \pageref*{cofibrant_operads} \linebreak
\noindent\hyperlink{objects}{$G$-objects}\dotfill \pageref*{objects} \linebreak
\noindent\hyperlink{coloured_operads}{Coloured operads}\dotfill \pageref*{coloured_operads} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{existence_by_coalgebra_intervals}{Existence by coalgebra intervals}\dotfill \pageref*{existence_by_coalgebra_intervals} \linebreak
\noindent\hyperlink{RectificationOfAlgebras}{Rectification of algebras}\dotfill \pageref*{RectificationOfAlgebras} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{monoids_associative_algebras}{Monoids (associative algebras)}\dotfill \pageref*{monoids_associative_algebras} \linebreak
\noindent\hyperlink{AInfAlgebras}{$A_\infty$-Algebras}\dotfill \pageref*{AInfAlgebras} \linebreak
\noindent\hyperlink{algebras_and_simplicial_lie_algebras}{$L_\infty$-algebras and simplicial Lie algebras}\dotfill \pageref*{algebras_and_simplicial_lie_algebras} \linebreak
\noindent\hyperlink{HomotopyCoherentDiagrams}{Homotopy coherent diagrams}\dotfill \pageref*{HomotopyCoherentDiagrams} \linebreak
\noindent\hyperlink{InfCatOfMods}{$(\infty,1)$-Categories of algebras and bimodules over an operad}\dotfill \pageref*{InfCatOfMods} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A [[model category]] structure on a [[category]] of [[algebras over an operad]] enriched in some suitable [[homotopical category]] $\mathcal{E}$ is supposed to be a [[presentable (infinity,1)-category|presentation]] of the [[(∞,1)-category]] of [[∞-algebras over an (∞,1)-operad]].

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

\hypertarget{existence}{}\subsection*{{Existence}}\label{existence}

\hypertarget{cofibrant_operads}{}\subsubsection*{{Cofibrant operads}}\label{cofibrant_operads}

\begin{utheorem}
Let $C$ be a [[cofibrantly generated model category|cofibrantly generated]] [[symmetric monoidal model category]]. Let $O$ be a [[cofibrant object|cofibrant]] [[operad]]. If $C$ satisfies the [[monoid axiom in a monoidal model category]], then there is an induced [[model structure]] on the category $Alg_C(O)$ of [[algebras over an operad]].

\end{utheorem}
See (\hyperlink{Spitzweck01}{Spitzweck 01}, Theorem 4).

\hypertarget{objects}{}\subsubsection*{{$G$-objects}}\label{objects}

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

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


\item a [[monoidal model category]];



\end{itemize}
such that

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


\item the tensor unit $I$ is cofibrant.



\end{itemize}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
Under these conditions there is for each [[finite group]] $G$ the structure of a [[monoidal model category]] on the category $\mathcal{E}^{\mathbf{B}G}$ of objects in $\mathcal{E}$ equipped with a $G$-[[action]], for which the [[forgetful functor]]

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

\end{prop}
This is discussed in the examples at \emph{[[monoidal model category]]}.

\hypertarget{coloured_operads}{}\subsubsection*{{Coloured operads}}\label{coloured_operads}

For $C \in$ [[Set]] a set of colours and $P$ a $C$-[[coloured operad]] in $\mathcal{E}$ we write $Alg_{\mathcal{E}}(P)$ for the category of $P$-[[algebras over an operad]]. There is a [[forgetful functor]]

\begin{displaymath}
U_P \;\colon\; Alg_{\mathcal{E}}(P) \to \mathcal{E}^C
\end{displaymath}
from the category of [[algebra over an operad|algebras over the operad]] in $\mathcal{E}$ to the underlying $C$-colored objects of $\mathcal{E}$.

\begin{defn}
\label{admissible}\hypertarget{admissible}{}
A $C$-[[coloured operad]] $P$ is called \textbf{admissible} if the [[transferred model structure]] on $Alg_{\mathcal{E}}(P)$ along the [[forgetful functor]]

\begin{displaymath}
U_P : Alg_{\mathcal{E}}(P)
  \to \mathcal{E}^{C}
\end{displaymath}
exists.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
So if $P$ is admissible, then $Alg_{\mathcal{E}}(P)$ carries the model structure where a morphism of $P$ algebras $f : A \to B$ is a fibration or weak equivalence if the underlying morphism in $\mathcal{E}$ is, respectively.

Below we discuss general properties of $P$ under which this model structure indeed exists.

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

\hypertarget{existence_by_coalgebra_intervals}{}\subsubsection*{{Existence by coalgebra intervals}}\label{existence_by_coalgebra_intervals}

The above [[transferred model structure]] on [[algebras over an operad]] exists if there is a suitable [[interval object]] in $\mathcal{E}$.

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{cocommutative coalgebra [[interval object]]} $H\in \mathcal{E}$ is

\begin{itemize}%
\item a cocommutative co-unital [[comonoid]] in $\mathcal{E}$


\item equipped with a factorization

\begin{displaymath}
\nabla : I \coprod I \hookrightarrow H \to I
\end{displaymath}
of the [[codiagonal]] on $I$ into two [[homomorphism]]s of comonoids with the first a cofibration and the second a weak equivalence in $\mathcal{E}$.



\end{itemize}
\end{defn}
\begin{example}
\label{}\hypertarget{}{}
Such cocommutative coalgebra intervals exist in

\begin{itemize}%
\item the [[model structure on topological spaces|model structure on compactly generated topological spaces]];


\item the [[model structure on simplicial sets]];


\item the [[model structure on symmetric spectra]].



\end{itemize}
In

\begin{itemize}%
\item the [[model structure on chain complexes]]

\end{itemize}
there is a coalgebra interval.

\end{example}
\begin{theorem}
\label{}\hypertarget{}{}
If $\mathcal{E}$ has a [[symmetric monoidal functor|symmetric monoidal]] fibrant replacement functor and a coalgebra [[interval object]] $H$ then every non-symmetric [[coloured operad]] in $\mathcal{E}$ is admissible, def. \ref{admissible}: the [[transferred model structure]] on algebras exists.

If the interval is moreover cocommutative, then the same is true for every symmetric coloured operad.

\end{theorem}
This is (\hyperlink{BergerMoerdiskAlgebras}{BergerMoerdijk, theorem 2.1}), following (\hyperlink{BergerMoerdijkHomotopy}{BergerMoerdijk-Homotopy, theorem 3.2}). For more details see at \emph{[[model structure on operads]]}.

\begin{remark}
\label{}\hypertarget{}{}
Since the coalgebra interval in the [[category of chain complexes]] is not cocommutative, this case requires special discussion, as some of the statements below will not apply to it. For more on this case see \emph{[[model structure on dg-algebras over an operad]]}.

\end{remark}
\hypertarget{RectificationOfAlgebras}{}\subsubsection*{{Rectification of algebras}}\label{RectificationOfAlgebras}

Recall the notion of resolutions of operads and of the \emph{[[Boardman-Vogt resolution]]} $W(H,P)$ from [[model structure on operads]].

We now discuss conditions under which model categories of algebras over a resolved operad is Quillen equivalent to that over the original operad. This yields general \emph{[[rectification]]} results for homotopy-algebras over an operad (see also the \emph{\hyperlink{Examples}{Examples}} below.)

\begin{theorem}
\label{}\hypertarget{}{}
Let $\mathcal{E}$ be in addition a [[left proper model category]].

Then for $\phi : P \to Q$ a weak equivalence between admissible $\Sigma$-cofibrant well-pointed $C$-coloured operads in $\mathcal{E}$, the [[adjunction]]

\begin{displaymath}
(\phi_! \dashv \phi^*) : 
  Alg_\mathcal{E}(P)
  \stackrel{\leftarrow}{\to}
  Alg_\mathcal{E}(Q)
\end{displaymath}
is a [[Quillen equivalence]].

\end{theorem}
This is (\hyperlink{BergerMoerdijkAlgebras}{BergerMoerdijk, theorem 4.1}).

\begin{theorem}
\label{RectificationTheorem}\hypertarget{RectificationTheorem}{}
\textbf{(rectification of homotopy $T$-algebras)}

Let still $\mathcal{E}$ be left proper.

Let $P$ be an admissible $\Sigma$-cofibrant operad in $\mathcal{E}$ such that also $W(H,P)$ is admissible.

Then

\begin{displaymath}
(\epsilon_! \dashv \epsilon^*) : 
  Alg_\mathcal{E}(P)
  \stackrel{\leftarrow}{\to}
  Alg_\mathcal{E}(W(H,P))
\end{displaymath}
is a [[Quillen equivalence]].

\end{theorem}
\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\hypertarget{monoids_associative_algebras}{}\subsubsection*{{Monoids (associative algebras)}}\label{monoids_associative_algebras}

For $P = Assoc$ the [[associative operad]] it category of algebras $Alg_{\mathcal{E}} P$ is the [[category of monoids]] in $\mathcal{E}$. The above model structure on $Alg_{\mathcal{E}} P$ is the standard [[model structure on monoids in a monoidal model category]].

\hypertarget{AInfAlgebras}{}\subsubsection*{{$A_\infty$-Algebras}}\label{AInfAlgebras}

Let $Assoc$ be the [[associative operad]] in [[Set]] regarded as an operad in [[Top]] under the [[discrete space]] embedding $Disc : Set \to Top$.

Let $I_*$ be the operad whose algebras are pointed objects. There is a canonical morphism $i : I_* \to Assoc$.

\begin{lemma}
\label{}\hypertarget{}{}
The [[relative Boardman-Vogt resolution]]

\begin{displaymath}
I_* \hookrightarrow I_*[i] \hookrightarrow W([0,1], I_* \to Assoc)  \stackrel{\simeq}{\to} Assoc
\end{displaymath}
produces precisely [[Jim Stasheff|Stasheff]]`s [[A-∞ operad]].

\end{lemma}
This is (\hyperlink{BergerMoerdijkAlgebras}{BergerMoerdijk, page 13})

\begin{cor}
\label{}\hypertarget{}{}
Every [[A-∞ space]] is equivalent as an $A_\infty$-space to a topological [[monoid]].

\end{cor}
\begin{proof}
This follows from the \hyperlink{RectificationTheorem}{rectification theorem}, using that by the above algebras over $W([0,1], I_* \to Assoc)$ are precisely [[A-∞ space]]s.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
This is a classical statement. See [[A-∞ algebra]] for background and references.

\end{remark}
\hypertarget{algebras_and_simplicial_lie_algebras}{}\subsubsection*{{$L_\infty$-algebras and simplicial Lie algebras}}\label{algebras_and_simplicial_lie_algebras}

Let $Lie$ be the [[Lie operad]].

A cofibrant resolution is $L_\infty$, the operad whose algebras in [[chain complex]]es are [[L-infinity algebras]].

Now (\ldots{})

\hypertarget{HomotopyCoherentDiagrams}{}\subsubsection*{{Homotopy coherent diagrams}}\label{HomotopyCoherentDiagrams}

Let $C$ be a small $\mathcal{E}$-[[enriched category]] with set of objects $Obj(C)$. There is an operad $Diag_{C}$

\begin{displaymath}
Diag_C(c_1, \cdots, c_n;c) = 
  \left\{
    \itexarray{
      C(c_1, c) & if n = 1
      \\
      \emptyset & otherwise
    }
  \right.
\end{displaymath}
whose algebras are [[enriched functor]]s

\begin{displaymath}
F : C \to \mathcal{E}
  \,,
\end{displaymath}
hence [[diagram]]s in $\mathcal{E}$. Then the [[Boardman-Vogt resolution]]

\begin{displaymath}
HoCoDiag_C := W(H,Diag_C)
\end{displaymath}
is the operad for [[homotopy coherent diagram]]s over $C$ in $\mathcal{E}$.

The [[rectification]] theorem above now says that every homotopy coherent diagram is equivalent to an ordinary $\mathcal{E}$-diagram. For $\mathcal{E} =$ [[Top]] this is known as [[Vogt's theorem]].

\hypertarget{InfCatOfMods}{}\subsubsection*{{$(\infty,1)$-Categories of algebras and bimodules over an operad}}\label{InfCatOfMods}

The constuction $Alg_{\mathcal{E}}(P)$ of a category of [[algebras over an operad]] is contravariantly functorial in $P$. Therefore if $P^\bullet$ is a [[cosimplicial object]] in the category of operads, we have that $Alg_{\mathcal{E}}(P^\bullet)$ is a (large) [[simplicial category]] of algebras. Moreover, the [[Boardman-Vogt resolution]] $W(P)$ is functorial in $P$.

These two facts together allow us to construct simplicial categories of homotopy algebras.

Specifically, there is a cosimplicial operad $Assoc^\bullet$ which

\begin{itemize}%
\item in degree 0 is the usual [[associative operad]] $Assoc^0  = Assoc$,


\item in degree 1 is the operad whose algebras are triples consisting of two associative monoids and one [[bimodule]] between them;


\item in degree 2 it is the operad whose algebras are tuples consisting of three associative algebras $A_0, A_1, A_2$ as well as one $A_i$-$A_j$-bimodule $N_{ i j}$ for each $0 \leq i \lt j \leq 2$ and a homomorphism of bimodules

\begin{displaymath}
N_{0 1} \otimes_{A_1} N_{1 2} \to N_{0 2}
\end{displaymath}

\item and so on.



\end{itemize}
The simplicial category of algebras over $Assoc^\bullet$ is one incarnation of the [[2-category]] of algebras, bimodules and bimodules homomorphisms.

We can pass to the corresponding $\infty$-algebras by applying the [[Boardman-Vogt resolution]] to the entire cosimplicial diagram of operads, to obtain the cosimplicial [[A-∞ operad]]

\begin{displaymath}
A_\infty^\bullet := W(Assoc^\bullet)
  \,.
\end{displaymath}
The [[simplicial category]] of algebras over this has as objects [[A-∞ algebra]]s, as morphism bimodules between these, and so on.

This is discussed in (\hyperlink{BergerMoerdijkAlgebras}{BergerMoerdijkAlgebras, section 6}).

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

\begin{itemize}%
\item [[model structure on monoids in a monoidal model category]]

\begin{itemize}%
\item [[model structure on dg-algebras]]


\item [[model structure for ring spectra]]



\end{itemize}

\item [[algebra over a monad]]

[[∞-algebra over an (∞,1)-monad]]

\begin{itemize}%
\item [[model structure on algebras over a monad]]

\end{itemize}

\item [[algebra over an algebraic theory]]

[[∞-algebra over an (∞,1)-algebraic theory]]

\begin{itemize}%
\item [[homotopy T-algebra]] / [[model structure on simplicial T-algebras]]

\end{itemize}

\item [[(∞,1)-operad]], [[model structure on operads]]

\begin{itemize}%
\item [[algebra over an (∞,1)-operad]], \textbf{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}


\end{itemize}


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

A general discussion of the [[model structure on operads]] is in

\begin{itemize}%
\item [[Clemens Berger]], [[Ieke Moerdijk]], \emph{Axiomatic homotopy theory for operads} Comment. Math. Helv. 78 (2003), 805--831. (\href{http://arxiv.org/abs/math/0206094}{arXiv:math/0206094})

\end{itemize}
See also

\begin{itemize}%
\item [[Markus Spitzweck]], \emph{Operads, Algebras and Modules in General Model Categories}, \href{http://arxiv.org/abs/math/0101102}{arXiv:math/0101102}.

\end{itemize}
The concrete construction of the specific cofibrant resolutions in these structures going by the name [[Boardman-Vogt resolution]] is in

\begin{itemize}%
\item [[Clemens Berger]], [[Ieke Moerdijk]], \emph{The Boardman-Vogt resolution of operads in monoidal model categories}, Topology 45 (2006), 807--849. (\href{http://math.unice.fr/~cberger/BV.pdf}{pdf})

\end{itemize}
The discussion of the model structure on algebras over a suitable operad is in

\begin{itemize}%
\item [[Clemens Berger]], [[Ieke Moerdijk]], \emph{Resolution of coloured operads and rectification of homotopy algebras} (\href{http://arxiv.org/abs/math/0512576}{arXiv:math/0512576})

\end{itemize}
More discussion on the transport of operad algebra structures along [[Quillen adjunctions]]/[[Bousfield localization of model categories|Bousfield localizations]] between the underlying model categories is in

\begin{itemize}%
\item [[Carles Casacuberta]], [[Javier Gutiérrez]], [[Ieke Moerdijk]], [[Rainer Vogt]], \emph{Localization of algebras over coloured operads}, Proceedings of the London Mathematical Society (3) 101 (2010), no. 1, 105-136 (\href{http://arxiv.org/abs/0806.3983}{arXiv:0806.3983})


\item [[Javier Gutiérrez]], \emph{Transfer of algebras over operads along derived Quillen adjunctions}, Journal of the London Mathematical Society 86 (2012), 607-625 (\href{http://arxiv.org/abs/1104.0584}{arXiv:1104.0584})



\end{itemize}
Discussion with an eye towards [[ring spectra]] realized as [[symmetric spectra]] is in

\begin{itemize}%
\item [[Stefan Schwede]], section III.6 of \emph{[[Symmetric spectra]]}, 2012 (\href{http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf}{pdf})

\end{itemize}
Discussion with application to [[homotopical algebraic quantum field theory]] is in

\begin{itemize}%
\item [[Marco Benini]], [[Alexander Schenkel]], [[Lukas Woike]], \emph{Homotopy theory of algebraic quantum field theories} (\href{https://arxiv.org/abs/1805.08795}{arXiv:1805.08795})

\end{itemize}
[[!redirects model structures on algebras over an operad]]

[[!redirects model category of algebras over an operad]] [[!redirects model categories of algebras over an operad]]



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