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\section*{(∞,1)-algebraic theory}

\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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{Models}{Models}\dotfill \pageref*{Models} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{simplicial_1algebras}{Simplicial 1-algebras}\dotfill \pageref*{simplicial_1algebras} \linebreak
\noindent\hyperlink{homotopy_algebras}{Homotopy $T$-algebras}\dotfill \pageref*{homotopy_algebras} \linebreak
\noindent\hyperlink{simplicial_theories}{Simplicial theories}\dotfill \pageref*{simplicial_theories} \linebreak
\noindent\hyperlink{structuresheaves}{Structure-$(\infty,1)$-sheaves}\dotfill \pageref*{structuresheaves} \linebreak
\noindent\hyperlink{EInfty}{Symmetric monoidal $(\infty,1)$-Categories and $E_\infty$-algebras}\dotfill \pageref*{EInfty} \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}

In as far as an [[algebraic theory]] or [[Lawvere theory]] is nothing but a [[small category]] with finite [[product]]s and an [[algebra]] for the theory a product-preserving [[functor]] to [[Set]], the notion has an evident generalization to [[higher category theory]] and in particular to [[(∞,1)-category]] theory.

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

\begin{defn}
\label{}\hypertarget{}{}
An \textbf{$(\infty,1)$-Lawvere theory} is (given by a syntactic $(\infty,1)$-category that is) an [[(∞,1)-category]] $C$ with finite [[limit in a quasi-category|(∞,1)-product]]s. An $(\infty,1)$-algebra for the theory is an [[(∞,1)-functor]] $C \to$ [[∞Grpd]] that preserves these products.

The $(\infty,1)$-category of [[∞-algebras over an (∞,1)-algebraic theory]] is the full [[sub-(∞,1)-category]]

\begin{displaymath}
Alg_{(\infty,1)}(C) \hookrightarrow PSh_{(\infty,1)}(C^{op})
\end{displaymath}
of the [[(∞,1)-category of (∞,1)-presheaves]] on $C^{op}$ on the product-preserving $(\infty,1)$-functors

\end{defn}
In a full $(\infty,1)$-category theoretic context this appears as [[Higher Topos Theory|HTT, def. 5.5.8.8]]. A definition in terms of [[simplicially enriched categories]] and the [[model structure on sSet-categories]] to present $(\infty,1)$-categories is in \href{(http://www.math.muni.cz/~rosicky/papers/infty6.pdf}{Ros}. The introduction of that article lists further and older occurences of this definition.

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

\begin{prop}
\label{}\hypertarget{}{}
Let $C$ be an [[(∞,1)-category]] with finite [[limit in a quasi-category|product]]s. Then

\begin{itemize}%
\item $Alg_{(\infty,1)}(C)$ is an [[accessible (∞,1)-functor|accessible]] [[localization of an (∞,1)-category|localization]] of the [[(∞,1)-category of (∞,1)-presheaves]] $PSh_{(\infty,1)}(C^{op})$ (on the [[opposite (∞,1)-category|opposite]]).

So in particular it is a [[locally presentable (∞,1)-category]].


\item $Alg_{(\inft)}$ is a [[compactly generated (∞,1)-category]].


\item The $(\infty,1)$-[[Yoneda embedding]] $j : C^{op} \to PSh_{(\infty,1)}(C^{op})$ factors through $Alg_{(\infty,1)}(C)$.


\item The full subcategory $Alg_{(\infty,1)}(C) \hookrightarrow PSh_{(\infty,1)}(C)$ is stable under [[sifted colimit]]s.



\end{itemize}
\end{prop}
This is [[Higher Topos Theory|HTT, prop. 5.5.8.10]].

\hypertarget{Models}{}\subsection*{{Models}}\label{Models}

There are various [[model category]] [[presentable (∞,1)-category|presentations]] of $Alg_{(\infty,1)}(C) \hookrightarrow PSh_{(\infty,1)}(C^{op})$.

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
  [T, sSet]^\circ
  \,,
\end{displaymath}
where we regard $T$ 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 $f : T \to \infty \mathrm{Grp}$ is equivalent to a \emph{rectified} on $F : T \to KanCplx$. 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 complex]]es.

If $T$ is an ordinary category with products, hence an ordinary [[Lawvere theory]], then such a functor is called a \textbf{[[homotopy T-algebra]]}. There is a model category structure on these (see there).

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 simplicial T-algebras]].

\begin{prop}
\label{}\hypertarget{}{}
Let $C$ be a [[category]] with finite [[product]]s, and let $sTAlg \subset Func(C,sSet)$ be the [[full subcategory]] of the [[functor category]] from $C$ to [[sSet]] on those functors that preserve these products.

Then $sAlg(C)$ carries the structure of a [[model category]] $sAlg(C)_{proj}$ where the weak equivalences and the fibrations are objectwise those in the standard [[model structure on simplicial sets]].

\end{prop}
This is due to (\hyperlink{Quillen}{Quillen}).

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

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

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

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

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

\hypertarget{simplicial_1algebras}{}\subsubsection*{{Simplicial 1-algebras}}\label{simplicial_1algebras}

For $T$ (the [[syntactic category]] of) an ordinary [[algebraic theory]] (a [[Lawvere theory]]) let $T Alg$ be the category of its ordinary algebras, the ordinary product-preserving functors $T \to Set$.

We may regard $T$ as an $(\infty,1)$-category and consider its $(\infty,1)$-algebras. By the above discussion, these are modeled by product-presering functors $T \to sSet$. But this are equivalently [[simplicial object]]s in $T$-algebras

\begin{displaymath}
[T, sSet]_\times \simeq T Alg^{\Delta^{op}}
  \,.
\end{displaymath}
There is a standard [[model structure on simplicial T-algebras]] and we find that simplicial $T$-1-algebras model $T$-$(\infty,1)$-algebras.

\hypertarget{homotopy_algebras}{}\subsubsection*{{Homotopy $T$-algebras}}\label{homotopy_algebras}

For $T$ an ordinary Lawvere theory, there is also a model category structure on ordinary functors $T \to sSet$ that preserve the products only up to weak equivalence. Such functors are called [[homotopy T-algebra]]s.

This model structure is equivalent to the [[model structure on simplicial T-algebras]] (see [[homotopy T-algebra]] for details) but has the advantage that it is a left [[proper model category]].

\hypertarget{simplicial_theories}{}\subsubsection*{{Simplicial theories}}\label{simplicial_theories}

There is a notion of \emph{simplicial algebraic theory} that captures some class of $(\infty,1)$-algebraic theories. For the moment see section 4 of (\hyperlink{Rezk}{Rezk})

\hypertarget{structuresheaves}{}\subsubsection*{{Structure-$(\infty,1)$-sheaves}}\label{structuresheaves}

A pre[[geometry (for structured (∞,1)-toposes)]] is a (multi-sorted) $(\infty,1)$-algebraic theory. A \emph{structure $(\infty,1)$-sheaf} on an [[(∞,1)-topos]] $\mathcal{X}$ in the sense of [[structured (∞,1)-topos]]es is an $\infty$-algebra over this theory

\begin{displaymath}
\mathcal{O} : \mathcal{T} \to \mathcal{X}
\end{displaymath}
in the $(\infty,1)$-topos $\mathcal{X}$ -- a special one satisfying extra conditions that make it indeed behave like a sheaf of \emph{function algebras} .

\hypertarget{EInfty}{}\subsubsection*{{Symmetric monoidal $(\infty,1)$-Categories and $E_\infty$-algebras}}\label{EInfty}

There is a $(2,1)$-algebraic theory whose algebras in [[(∞,1)Cat]] are [[symmetric monoidal (∞,1)-categories]]. Hence monoids in these algebras are [[E-∞ algebra]]s (see [[monoid in a monoidal (∞,1)-category]]).

This is in (\hyperlink{Cranch}{Cranch}). For more details see [[(2,1)-algebraic theory of E-infinity algebras]].

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

\begin{itemize}%
\item [[algebraic theory]] / [[Lawvere theory]] / [[essentially algebraic theory]]

\begin{itemize}%
\item [[2-Lawvere theory]]


\item \textbf{algebraic $(\infty,1)$-theory} / [[essentially algebraic (∞,1)-theory]]

\begin{itemize}%
\item [[simplicial Lawvere theory]]

\end{itemize}


\end{itemize}

\item [[monad]] / [[(∞,1)-monad]]


\item [[operad]] / [[(∞,1)-operad]]



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

The model structure presentation for the $(\infty,1)$-category of $(\infty,1)$-algebras goes back all the way to

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

\end{itemize}
A characterization of $(\infty,1)$-categories of $(\infty,1)$-algebras in terms of [[sifted colimit]]s is given in

\begin{itemize}%
\item J. Rosicky \emph{On homotopy varieties} (\href{http://www.math.muni.cz/~rosicky/papers/infty6.pdf}{pdf})

\end{itemize}
using the incarnation of $(\infty,1)$-categories as [[simplicially enriched categories]].

An $(\infty,1)$-categorical perspective on these homotopy-algebraic theories is given in

\begin{itemize}%
\item [[Andre Joyal]], \emph{The theory of quasi-categories and its applications}, lectures at CRM Barcelona February 2008, draft \href{http://www.crm.cat/HigherCategories/hc2.pdf#page=44}{hc2.pdf}\_

\end{itemize}
from page 44 on.

A detailed account in the context of a general theory of [[(∞,1)-category of (∞,1)-presheaves]] is the context of section 5.5.8 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Topos Theory]]} .

\end{itemize}
The [[model category]] [[presentable (infinity,1)-category|presentations]] of $(\infty,1)$-algebras is studied 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}
where it is shown that every such model is [[Quillen equivalence|Quillen equivalent]] to a left [[proper model category]]. The article uses a monadic definition of $(\infty,1)$-algebras.

A discussion of [[homotopy T-algebra]]s and their strictification 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 for multi-sorted theories in

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

\end{itemize}
A discussion of [[E-∞ algebra]]-structures in terms of $(\infty,1)$-algebraic theories is in

\begin{itemize}%
\item [[James Cranch]], \emph{Algebraic Theories and $(\infty,1)$-Categories} (\href{http://arxiv.org/abs/1011.3243}{arXiv})

\end{itemize}
See also

\begin{itemize}%
\item \href{http://mathoverflow.net/questions/118500/what-is-a-simplicial-commutative-ring-from-the-point-of-view-of-homotopy-theory}{MO discussion}

\end{itemize}
[[!redirects (infinity,1)-algebraic theory]] [[!redirects (infinity,1)-algebraic theories]] [[!redirects (∞,1)-algebraic theories]]

[[!redirects algebraic (∞,1)-theory]] [[!redirects algebraic (∞,1)-theories]]



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