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\begin{document}

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\section*{homotopy T-algebra}

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

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

[[!include higher algebra - contents]]

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

[[!include homotopy - 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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 \emph{homotopy $T$-algebra} over a [[Lawvere theory]] $T$ is a model for an $\infty$-algebra over $T$, when the latter is regarded as an [[(∞,1)-algebraic theory]].

As a model, homotopy $T$-algebras are equivalent to strict [[simplicial algebra]]s.

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

For $T$ (the syntactic category of) a [[Lawvere theory]] with generating object $x$ an ordinary [[algebra over a Lawvere theory]] [[functor]] $T \to Set$ that preserves products, in that for all $n \in \mathbb{N}$ the canonical morphism

\begin{displaymath}
\prod_{i = 1}^n A(p_i) :  A(x^n) \to (A(x))^n
\end{displaymath}
is an [[isomorphism]].

\begin{udefn}
A \textbf{homotopy $T$-algebra} is a functor $A : T \to$ [[sSet]] with values in [[Kan complex]]es such that for all $n \in \mathbb{N}$ this canonical morphism is a [[weak homotopy equivalence]].

\end{udefn}
For $n \in \mathbb{N}$ write $F_T(n)$ for the free simplicial $T$-algebra on $n$-generators, which is the image of $x^n$ under the [[Yoneda embedding]] $j : T^{op} \to [T,sSet]$. (See [[Lawvere theory]] for more on this.)

\begin{uprop}
A homotopy $T$-algebra is precisely

\begin{itemize}%
\item a fibrant object in the projective [[model structure on simplicial presheaves]];


\item which is a [[local object]] with respect to the canonical morphisms

\begin{displaymath}
\coprod F_T(1) \to F_T(n)
\end{displaymath}
for all $n \in \mathbb{N}$.



\end{itemize}
\end{uprop}
\begin{proof}
The fibrant objects in $[T,sSet]_{proj}$ are precisely the [[Kan complex]]-valued co-presheaves. Because $F_T(n)$ is [[representable functor|representable]], it is cofibrant in $[T,sSet]_{proj}$ (as one easily checks). Therefore the [[derived hom-space]]s between $F_T(\cdots)$ and a degreewise Kan complex-valued $A$ may be computed simply as the [[sSet]]-[[hom-object]]s of the [[simplicial model category]] $[T,sSet]$ and so the degreewise fibrant $A$ being a [[local object]] means that all morphisms of [[sSet]]-[[hom-object]]s

\begin{displaymath}
[T,sSet](F_T(n),A) \to [T,sSet](\coprod_n F_T(1), A)
  \,.
\end{displaymath}
Due to the respect of the [[hom-functor]] for [[limit]]s the expression on the right is

\begin{displaymath}
\cdots = \prod_n [T,sSet](F_T(1), A)
  \,.
\end{displaymath}
Using the [[Yoneda lemma]] the morphism in question is indeed isomorphic to

\begin{displaymath}
A(x^n) \to A(x)^n
  \,.
\end{displaymath}
\end{proof}
This observation motivated the following definition.

\begin{udef}
The \textbf{[[model category]] structure for homotopy $T$-algebras} is the left [[Bousfield localization of model categories|Bousfield localization]] $[T,sSet]_{proj,loc}$ of the projective [[model structure on simplicial presheaves]] $[T,sSet]_{proj}$ at the set of morphisms $\{\coprod_n F_T(1) \to F_T(b)\}_{n \in \mathbb{N}}$.

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

\begin{uprop}
The model structure for homotopy $T$-algebra $[T,sSet]_{proj,loc}$ is a [[left proper model category|left proper]] [[simplicial model category]].

\end{uprop}
\begin{proof}
Because the [[model structure on simplicial presheaves]] is and left [[Bousfield localization of model categories]] preserves these properties.

\end{proof}
\begin{ulemma}
The inclusion

\begin{displaymath}
i : T Alg^{\Delta^{op}} \hookrightarrow [T,sSet]
\end{displaymath}
has a [[left adjoint]]

\begin{displaymath}
F : [T,sSet] \to T Alg^{\Delta^{op}}
\end{displaymath}
\end{ulemma}
\begin{proof}
The [[limit]]s in $T Alg$ are easily seen to be limits in the underlying sets. Hence $i$ preserves all limits. The statement then follows by observing that the assumptions of the special [[adjoint functor theorem]] are met:

\begin{itemize}%
\item $T Alg$ is complete;


\item it is a [[well powered category]] since $[T,Set]$ is and the subobject in $T Alg$ are special subobjects in $[T,Set]$;


\item it has a small [[cogenerating set]] given by the representables.



\end{itemize}
\end{proof}
\begin{uremark}
An explicit description of $F$ is around [[Higher Topos Theory|HTT, lemma 5.5.9.5]].

\end{uremark}
\begin{utheorem}
Let $T Alg^{\Delta^{op}}_{proj}$ be the category of [[simplicial object|simplicial]] [[T-algebra]]s equipped with the standard [[model structure on simplicial algebras]] (with weak equivalences and fibrations the degreewise weak equivalences and fibrations in simplicial sets).

The adjunction from the previous lemma

\begin{displaymath}
T Alg^{\Delta^{op}} \stackrel{\overset{F}{\leftarrow}}{\hookrightarrow}
  [T,sSet] = [T,Set]^{\Delta^{op}}
\end{displaymath}
is a [[Quillen adjunction]] which is a [[Quillen equivalence]]

\begin{displaymath}
T Alg^{\Delta^{op}}_{proj} \simeq
  [T,sSet]_{proj,loc}
  \,.
\end{displaymath}
\end{utheorem}
This is theorem 1.3 in (\hyperlink{Badzioch}{Badzioch})

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

The model structure on homotopy $T$-algebras for $T =$ [[CartSp]] the Lawvere theory of [[smooth algebra]]s is considered in (\hyperlink{Spivak}{Spivak}) in the study of [[derived smooth manifold]]. (There is also a bit of disucssion of the relation to the model structure on simplicial algebras there.)

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

\begin{itemize}%
\item [[algebra over a monad]]

[[∞-algebra over an (∞,1)-monad]]


\item [[algebra over an algebraic theory]]

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

\begin{itemize}%
\item \textbf{homotopy T-algebra} / [[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}

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}
the model structure on homotopy $T$-algebras is discussed and its Quillen equivalence to simplicial $T$-algebras is proven.

A related discussion showing that simplicial $T$ algebras model all $\infty$-$T$-algebras is in

\begin{itemize}%
\item [[Julie Bergner]], \emph{Rigidification of algebras over multi-sorted theories} , Algebraic and Geometric Topoogy 7, 2007.

\end{itemize}
The model structure on homotopy $T$-algebras for $T =$ [[CartSp]] the Lawvere theory of [[smooth algebra]]s is considered in

\begin{itemize}%
\item [[David Spivak]], \emph{Derived smooth manifolds} (\href{http://arxiv.org/abs/0810.5174}{arXiv:0810.5174})

\end{itemize}
[[!redirects homotopy T-algebras]]

[[!redirects model structure on homotopy T-algebras]]



\end{document}