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\begin{document}

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\section*{model structure on simplicial algebras}

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

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

[[!include model category theory - contents]]

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

[[!include homotopy - 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{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{OverOrdinaryLawvereTheories}{For algebras over ordinary Lawvere theories}\dotfill \pageref*{OverOrdinaryLawvereTheories} \linebreak
\noindent\hyperlink{ForAlgebrasOverSimplicialLawvereTheories}{For algebras over simplicial Lawvere theories}\dotfill \pageref*{ForAlgebrasOverSimplicialLawvereTheories} \linebreak
\noindent\hyperlink{OtherCases}{Other cases}\dotfill \pageref*{OtherCases} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_homotopy_algebras}{Relation to homotopy $T$-algebras}\dotfill \pageref*{relation_to_homotopy_algebras} \linebreak
\noindent\hyperlink{homotopy_groups}{Homotopy groups}\dotfill \pageref*{homotopy_groups} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{simplicial_associative_algebras}{Simplicial associative $k$-algebras}\dotfill \pageref*{simplicial_associative_algebras} \linebreak
\noindent\hyperlink{simplicial_commutative_algebras}{Simplicial commutative $k$-algebras}\dotfill \pageref*{simplicial_commutative_algebras} \linebreak
\noindent\hyperlink{simplicial_lie_algebras}{Simplicial Lie algebras}\dotfill \pageref*{simplicial_lie_algebras} \linebreak
\noindent\hyperlink{SimplicialCommutativeHopfAlgebras}{Simplicial complete Hopf algebras}\dotfill \pageref*{SimplicialCommutativeHopfAlgebras} \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}

For $T$ a [[Lawvere theory]] and $T Alg$ the [[category]] of [[algebra over a Lawvere theory]], there is a [[model category]] structure on the category $T Alg^{\Delta^{op}}$ of [[simplicial object|simplicial]] $T$-algebras which models the $\infty$-algebras for $T$ regarded as an [[(∞,1)-algebraic theory]].

\hypertarget{details}{}\subsection*{{Details}}\label{details}

First we consider the case of [[simplicial objects]] in algebras over an ordinary Lawvere theory:

\begin{itemize}%
\item \emph{\hyperlink{OverOrdinaryLawvereTheories}{For algebras over ordinary Lawvere theories}}

\end{itemize}
Then we generalize to the case that the Lawvere theory itself is simplicial:

\begin{itemize}%
\item \emph{\hyperlink{ForAlgebrasOverSimplicialLawvereTheories}{For algebras over simplicial Lawvere theories}}

\end{itemize}
\hypertarget{OverOrdinaryLawvereTheories}{}\subsubsection*{{For algebras over ordinary Lawvere theories}}\label{OverOrdinaryLawvereTheories}

Recall that the [[(∞,1)-category of (∞,1)-presheaves]] $PSh_{(\infty,1)}(C^{op})$ itself is modeled by the [[model structure on simplicial presheaves]]

\begin{displaymath}
PSh_{(\infty,1)}(C^{op})
  \simeq
  [C, sSet]^\circ
  \,,
\end{displaymath}
where we regard $C$ as a [[Kan complex]]-[[enriched category]] and have on the right the [[sSet]]-[[enriched functor category]] with the projective or injective model structure, and $(-)^\circ$ denoting the full enriched subcategory on fibrant-cofibrant objects.

This says in particular that every weak $(\infty,1)$-functor $f : C \to \infty \mathrm{Grp}$ is equivalent to a \emph{rectified} one $F : C \to KanCplx$. And $f \in PSh_{(\infty,1)}(C^{op})$ belongs to $Alg_{(\infty,1)}(C)$ if $F$ preserves finite products \emph{weakly} in that for $\{c_i \in C\}$ a finite collection of objects, the canonical natural morphism

\begin{displaymath}
F(c_1 \times \cdots, \c_n)
  \to
  F(c_1) \times \cdots \times F(c_n)
\end{displaymath}
is a [[homotopy equivalence]] of [[Kan complexes]].

We now look at model category structure on \emph{strictly} product preserving functors $C \to sSet$, which gives an equivalent model for $Alg_{(\infty,1)}(C)$. See [[model structure on homotopy T-algebras]].

\begin{prop}
\label{ModelStructureOnSimplicialAlgebrasOverOrdinaryLawveretheory}\hypertarget{ModelStructureOnSimplicialAlgebrasOverOrdinaryLawveretheory}{}
Let $T$ be the [[syntactic category]] of a [[Lawvere theory]], and let $T Alg^{\Delta^{op}} \subset Func(T,sSet)$ be the [[full subcategory]] of the [[functor category]] from $T$ to [[sSet]] on those functors that preserve these [[finite products]].

Then $T Alg^{\Delta^{op}}$ carries the structure of a [[model category]] $(T Alg^{\Delta^{op}})_{proj}$ where the weak equivalences and the fibrations are those maps underlying which are weak equivalences or fibrations, respectively, in the [[classical model structure on simplicial sets]].

\end{prop}
This is due to (\hyperlink{Quillen67}{Quillen 67, II.4 theorem 4}).

The inclusion $i : sAlg(C) \hookrightarrow sPSh(C^{op})_{proj}$ into the projective [[model structure on simplicial presheaves]] evidently preserves fibrations and acyclic fibrations and gives a [[Quillen adjunction]]

\begin{displaymath}
sAlg(C)_{proj}
  \stackrel{\leftarrow}{\underset{i}{\hookrightarrow}}
  sPSh(C^{op})
  \,.
\end{displaymath}
\begin{prop}
\label{}\hypertarget{}{}
The total right [[derived functor]]

\begin{displaymath}
\mathbb{R}i \;\colon\; Ho(sAlg(C)_{proj}) \to Ho(sPSh(C^{op})_{proj})
\end{displaymath}
is a [[full and faithful functor]] and an object $F \in sPSh(C^{op})$ belongs to the [[essential image]] of $\mathbb{R}i$ precisely if it preserves products up to [[weak homotopy equivalence]].

\end{prop}
This is due to (\hyperlink{Badzioch02}{Badzioch 02, def. 5.2, prop. 5.4, cor. 5.7}).

It follows that the natural $(\infty,1)$-functor

\begin{displaymath}
(sAlg(C)_{proj})^\circ 
    \stackrel{}{\longrightarrow}
  PSh_{(\infty,1)}(C^{op})
\end{displaymath}
is an [[equivalence of quasi-categories|equivalence]].

\hypertarget{ForAlgebrasOverSimplicialLawvereTheories}{}\subsubsection*{{For algebras over simplicial Lawvere theories}}\label{ForAlgebrasOverSimplicialLawvereTheories}

\begin{defn}
\label{SimplicialLawvereTheory}\hypertarget{SimplicialLawvereTheory}{}
Let $\Gamma = (Skel(FinSet^{\ast/}))$ be [[Segal's category]], the [[opposite category]] of a [[skeleton]] of [[finite set|finite]] [[pointed sets]].

A \emph{[[simplicial Lawvere theory]]} is a a pointed [[simplicial category]] $T$ equipped with a [[functor]] $i \;\colon\;\Gamma \to T$ such that

\begin{enumerate}%
\item $T$ has the same [[set]] of [[objects]] as $\Gamma$;


\item $i$ is the identity on objects


\item $i$ preserves [[finite products]]



\end{enumerate}
Given a simplicial theory $T$, then a \emph{simplicial $T$-algebra} is a product preserving [[simplicial functor]] $X$ to the [[simplicial category]] of [[pointed simplicial sets]]. The simplicial set

\begin{displaymath}
X(1_+) \in sSet
\end{displaymath}
(the value on the pointed 2-element set) is called the \emph{underlying simplicial set} of the $T$-algebra.

A [[homomorphism]] of $T$-algebras is a simplicial [[natural transformation]] between such functors. Write

\begin{displaymath}
T Alg \in sSet Cat
\end{displaymath}
for the resulting [[simplicial category]].

A homomorphism is called a \emph{weak equivalence} or a \emph{fibration} if on underlying simplicial sets it is a weak equivalence or fibration, respectively, in the [[classical model structure on simplicial sets]]. Write

\begin{displaymath}
(T Alg)_{proj}
\end{displaymath}
for the category equipped with these [[classes]] of morphisms.

\end{defn}
(\hyperlink{Schwede01}{Schwede 01, def. 2.1 and def. 2.2 and beginning of section 3})

\begin{remark}
\label{}\hypertarget{}{}
In the case that the [[simplicial Lawvere theory]] $T$ happens to be simplicially discrete, i.e. an ordinary [[category]], then the category $(T Alg)_{proj}$ of simplicial $T$-algebras from def. \ref{SimplicialLawvereTheory} with that from prop. \ref{ModelStructureOnSimplicialAlgebrasOverOrdinaryLawveretheory}.

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
For $T$ a [[simplicial Lawvere theory]] (def. \ref{SimplicialLawvereTheory}) the category $(T Alg)_{poj}$ from def. \ref{SimplicialLawvereTheory} is a [[simplicial model category]].

\end{prop}
This is due to (\hyperlink{Reedy74}{Reedy 74, theorem I}), reviewed in (\hyperlink{Schwede01}{Schwede 01}).

The analogous statement with the [[classical model structure on simplicial sets]] replaced by the [[classical model structure on topological spaces]] is due to (\hyperlink{Schw&#228;nzlVogt91}{Schw\"a{}nzl-Vogt 91}9

A comprehensive statement of these facts is in [[Higher Topos Theory|HTT, section 5.5.9]].

\hypertarget{OtherCases}{}\subsubsection*{{Other cases}}\label{OtherCases}

\begin{prop}
\label{QuillenCriterion}\hypertarget{QuillenCriterion}{}
Let $\mathcal{C}$ be a [[category]] with [[enough projectives]]. Say that a morphism $f$ in the category $\mathcal{C}^{\Delta^{\mathrm{op}}}$ of [[simplicial objects]] in $\mathcal{C}$ is a \emph{weak equivalence} or \emph{fibration} if for every [[projective object]] $P \in \mathcal{C}$ then $Hom(P,f)$ is a weak equivalence or fibration, respectively, in the [[classical model structure on simplicial sets]] (i.e. a [[Kan fibration]] or [[weak homotopy equivalence]], respectively). If now at least one of the following conditions is satisfied,

\begin{itemize}%
\item every object of $\mathcal{C}^{\Delta^{op}}$ is fibrant;


\item $\mathcal{C}$ is closed under [[colimits]] and has a [[small set]] of [[projective object|projective]] [[generators]];



\end{itemize}
then this makes $\mathcal{C}^{\Delta^{op}}$ a [[simplicial model category]].

\end{prop}
(\hyperlink{Quillen67}{Quillen 67, II.4, theorem 4})

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

\hypertarget{relation_to_homotopy_algebras}{}\subsubsection*{{Relation to homotopy $T$-algebras}}\label{relation_to_homotopy_algebras}

\begin{theorem}
\label{}\hypertarget{}{}
There is a [[Quillen equivalence]] between the model structure on simplicial $T$-algebras and the model structure for [[homotopy T-algebra]]s. (See there).

\end{theorem}
This is (\hyperlink{Badzioch02}{Badzioch 02, theorem 1.3}).

\hypertarget{homotopy_groups}{}\subsubsection*{{Homotopy groups}}\label{homotopy_groups}

Let $T$ be an \emph{abelian Lawvere theory}, a theory that contains the theory of [[abelian group]], $Ab \to T$. Then every simplicial $T$-algebra has an underlying abelian [[simplicial group]] and is necessarily a [[Kan complex]].

\begin{prop}
\label{}\hypertarget{}{}
The [[homotopy group]]s $\pi_*$ of a simplicial abelian $T$-agebra form an $\mathbb{N}$-graded $T$-algebra $\pi_*(A)$

\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
The inclusion of the full [[subcategory]] $i : T Alg \hookrightarrow T Alg^{\Delta^{op}}$ of ordinary $T$-algebra as the simplicially constant ones constitutes a [[Quillen adjunction]]

\begin{displaymath}
(\pi_0  \dashv i) : T Alg \stackrel{\overset{\pi_0}{\leftarrow}}{\underset{i}{\hookrightarrow}}
  T Alg^{\Delta^{op}}
\end{displaymath}
from the [[trivial model structure]] on $T Alg$.

The [[derived functor]] $\mathbb{R} i : T Alg \to Ho(T Alg^{\Delta^{op}})$ is a [[full and faithful functor]].

\end{prop}
This allows us to think of ordinary $T$-algebras a sitting inside $\infty$-$T$-algebras.

\begin{quote}%
all this is certainly true for ordinary $k$-algebras. Need to spell out general proof.


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

\hypertarget{simplicial_associative_algebras}{}\subsubsection*{{Simplicial associative $k$-algebras}}\label{simplicial_associative_algebras}

A [[simplicial ring]]s is a simplicial $T$-algebras for $T$ the Lawvere theory of rings.

Let $k$ be an ordinary commutative ring and $T$ the theory of commutative [[associative algebras]] over $k$. We write $T Alg$ as $sCAlg_k$ or $CAlg_k^{op}$.

Such simplicial $k$-algebras are discussed for instance in (\hyperlink{ToenVezzosi}{To\"e{}n-Vezzosi, section 2.2.1}). According to (\hyperlink{Schwede97}{Schwede 97}, Lemma 3.1.3), this model structure is [[proper model category|proper]].

\hypertarget{simplicial_commutative_algebras}{}\subsubsection*{{Simplicial commutative $k$-algebras}}\label{simplicial_commutative_algebras}

The following is a variant of the model structure on simplicial commutative algebras that is implied by the above general theorem.

\begin{prop}
\label{}\hypertarget{}{}
\textbf{(second model structure)}

Let $T$ be the [[Lawvere theory]] for ([[associative algebra|associative]] and) [[commutative algebras]] over a [[ring]] $k$. Then $(cAlg_k)^{\Delta^{op}}$ becomes a [[simplicial model category]] with

\begin{itemize}%
\item weak equivalences the morphisms whose underlying morphism of simplicial sets are weak equivalences in the [[classical model structure on simplicial sets]];


\item fibrations the morphisms $X \to Y$ such that $X \to \pi_0 X \times_{\pi_0 Y} Y$ is a degreewise surjection.



\end{itemize}
\end{prop}
This appears as (\hyperlink{GoerssSchemmerhorn}{Goerss-Schemmerhorn, theorem 4.17}).

\hypertarget{simplicial_lie_algebras}{}\subsubsection*{{Simplicial Lie algebras}}\label{simplicial_lie_algebras}

See at \emph{[[model structure on simplicial Lie algebras]]}.

\hypertarget{SimplicialCommutativeHopfAlgebras}{}\subsubsection*{{Simplicial complete Hopf algebras}}\label{SimplicialCommutativeHopfAlgebras}

Complete rational [[Hopf algebras]] are not the models of a [[Lawvere theory]], even though in some sense they are close (\hyperlink{Quillen67}{Quillen 67, bottom of p. 265 (61 of 92)}).

But they do satisfy the assumptions of proposition \ref{QuillenCriterion} (\hyperlink{Quillen69}{Quillen 69, appendix B, prop. 2.24}). Hence simplicial rational complete Hopf algebras form a [[simplicial model category]].

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

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


\item [[model structure on monoids]]


\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]] / \textbf{model structure on simplicial T-algebras}

\end{itemize}

\item [[algebra over an operad]]

[[∞-algebra over an (∞,1)-operad]]

\begin{itemize}%
\item [[model structure on algebras over an operad]]

\end{itemize}


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

The classical reference for the [[transferred model structure]] on simplicial $T$-algebras is

\begin{itemize}%
\item [[Dan Quillen]], \emph{Homotopical Algebra}, Lectures Notes in Mathematics 43, Springer Verlag, Berlin, (1967)

\end{itemize}
with some extra remarks in

\begin{itemize}%
\item [[Dan Quillen]], \emph{Rational homotopy theory}, The Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 (\href{http://www.jstor.org/stable/1970725}{JSTOR}, \href{http://www.math.northwestern.edu/~konter/gtrs/rational.pdf}{pdf})

\end{itemize}
The generalization to the case that the theory $T$ itself is allowed to be simplicial is due to

\begin{itemize}%
\item [[Christopher Reedy]], \emph{Homology of algebraic theories}, Ph.D. Thesis, University of California, San Diego, 1974

\end{itemize}
and the topological version is due to

\begin{itemize}%
\item [[Roland Schwänzl]], [[Rainer Vogt]], \emph{The categories of $A_\infty$- and $E_\infty$-monoids and ring spaces as closed simplicial and topological model categories}, Archives of Mathematics 56 (1991) 405-411

\end{itemize}
More is in

\begin{itemize}%
\item [[Stefan Schwede]], \emph{Stable homotopy of algebraic theories}, Topology 40 (2001) 1-41 (\href{http://www.math.uni-bonn.de/people/schwede/stable.pdf}{pdf})

\end{itemize}
In

\begin{itemize}%
\item [[Charles Rezk]], \emph{Every homotopy theory of simplicial algebras admits a proper model} (\href{http://arxiv.org/abs/math/0003065}{math/0003065}) ,

\end{itemize}
it is discussed that every model category of simplicial $T$-algebras is [[Quillen equivalence|Quillen equivalent]] to a left [[proper model category]].

The fact that the model structure on simplicial $T$-algebras serves to model $\infty$-algebras is in

\begin{itemize}%
\item [[Bernard Badzioch]], \emph{Algebraic theories in homotopy theory}, Annals of Mathematics, 155 (2002), 895-913 (\href{http://www.jstor.org/stable/3062135}{JSTOR})

\end{itemize}
and the multi-sorted version is in

\begin{itemize}%
\item [[Julie Bergner]], \emph{Rigidification of algebras over multi-sorted theories}, Algebraic and Geometric Topoogy 7 (2007) (\href{https://arxiv.org/abs/math/0508152}{arXiv:0508152})

\end{itemize}
The simplicial model structure on ordinary simplicial algebras (i.e. simplicial [[associative algebras]]) is discussed in

\begin{itemize}%
\item [[Paul Goerss]], [[Kirsten Schemmerhorn]], \emph{Model categories and simplicial methods} (\href{http://www.math.northwestern.edu/~pgoerss/papers/ucnotes.pdf}{pdf})


\item [[Stefan Schwede]], \emph{Spectra in model categories and applications to the algebraic cotangent complex}, Journal of Pure and Applied Algebra 120 (1997), pp. 77-104, \href{http://www.math.uni-bonn.de/people/schwede/modelspec.pdf}{pdf}.



\end{itemize}
Discussion of simplicial commutative associative algbras over a ring in the context of [[derived geometry]] is in

\begin{itemize}%
\item [[Bertrand Toën]], [[Gabriele Vezzosi]], \emph{Homotopical Algebraic Geometry II: geometric stacks and applications} (\href{http://arxiv.org/abs/math/0404373}{arXiv})

\end{itemize}
Discussion of divided [[power operations]] on simplicial algebras is in

\begin{itemize}%
\item [[Benoit Fresse]], \emph{On the homotopy of simplicial algebras over an operad}, Transactions of the AMS, volume 352, number 9 (\href{http://www.jstor.org/stable/118177}{jstor})

\end{itemize}
[[!redirects model structure on simplicial T-algebras]]

[[!redirects simplicial T-algebra]]

[[!redirects simplicial algebra]] [[!redirects simplicial algebras]]

[[!redirects model structure on simplicial abelian groups]]



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