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\begin{document}

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

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

\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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{MonadicityTheorem}{Barr-Beck monadicity theorem}\dotfill \pageref*{MonadicityTheorem} \linebreak
\noindent\hyperlink{HomotopyCoherence}{Homotopy coherence}\dotfill \pageref*{HomotopyCoherence} \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 $(\infty,1)$-monad is the [[vertical categorification]] of that of [[monad]] from the context of [[category|categories]] to that of [[(∞,1)-category|(∞,1)-categories]].

They relate to [[(∞,1)-adjunctions]] as [[monads]] relate to [[adjunctions]].

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

\hypertarget{MonadicityTheorem}{}\subsubsection*{{Barr-Beck monadicity theorem}}\label{MonadicityTheorem}

\begin{prop}
\label{CanonicalMonadicAdjunction}\hypertarget{CanonicalMonadicAdjunction}{}
Given an [[(∞,1)-monad]] $T$ on an [[(∞,1)-category]] $\mathcal{C}$, there is an [[(∞,1)-adjunction]]

\begin{displaymath}
(F \dashv U)
  \;\colon\;
  Alg_{\mathcal{C}}(T)
  \stackrel{\overset{F}{\leftrightarrow}}{\underset{U}{\longrightarrow}}
  \mathcal{C}
  \,,
\end{displaymath}
where $Alg_{\mathcal{C}}(T)$ is the (Eilenberg-Moore) [[(∞,1)-category of algebras over an (∞,1)-monad|(∞,1)-category of algebras over the (∞,1)-monad]] and where $U$ is the [[forgetful functor]] that remembers the underlying [[object]] of $\mathcal{C}$.

\end{prop}
This appears in (\hyperlink{RiehlVerity13}{Riehl-Verity 13, def. 6.1.15}).

The following is the refinement to [[(∞,1)-category theory]] of the classical [[Barr-Beck monadicity theorem]] which states sufficient conditions for recognizing an [[(∞,1)-adjunction]] as being canonically [[equivalence|equivalent]] to the one in prop. \ref{CanonicalMonadicAdjunction}, hence to be a \emph{[[monadic adjunction]]}.

\begin{theorem}
\label{InfinityBarrBeckTheorem}\hypertarget{InfinityBarrBeckTheorem}{}
Let $(L \dashv R)$ a pair of [[adjoint (∞,1)-functors]] such that

\begin{enumerate}%
\item $R$ is a [[conservative (∞,1)-functor]];


\item the [[domain]] [[(∞,1)-category]] of $R$ admits [[geometric realization]] ([[(∞,1)-colimit]]) of [[simplicial objects in an (∞,1)-category|simplicial objects]];


\item and $R$ preserves these



\end{enumerate}
then for $T \coloneqq R \circ L$ the essentially unique $(\infty,1)$-monad structure on the composite endofunctor, there is an [[equivalence of (∞,1)-categories]] identifying the [[domain]] of $R$ with the [[(∞,1)-category of algebras over an (∞,1)-monad]] $Alg_{\mathcal{C}}(T)$ over $T$ and $R$ itself as the canonical [[forgetful functor]] $U$ from prop. \ref{CanonicalMonadicAdjunction}.

\end{theorem}
This appears as ([[Higher Algebra|Higher Algebra, theorem 6.2.0.6, theorem 6.2.2.5]], \hyperlink{RiehlVerity13}{Riehl-Verity 13, section 7})

\hypertarget{HomotopyCoherence}{}\subsubsection*{{Homotopy coherence}}\label{HomotopyCoherence}

\begin{remark}
\label{}\hypertarget{}{}
An [[(∞,1)-adjunction]] $(L \dashv R) \colon \mathcal{C} \leftrightarrow \mathcal{D}$ is uniquely determined already by its image in the [[homotopy 2-category]] (\hyperlink{RiehlVerity13}{Riehl-Verity 13, theorem 5.4.14}). This is not in general true for $(\infty,1)$-monads $T \colon \mathcal{C} \to \mathcal{C}$. As these are [[monoids in an (∞,1)-category]] of [[endomorphisms]], they in general have relevant [[coherence]] data all the way up in degree. However, by the previous statement and the monadicity theorem \ref{InfinityBarrBeckTheorem}, for $(\infty,1)$-monads given via specified [[(∞,1)-adjunctions]] as $T \simeq R \circ L$ are determined by less (further) coherence data ([[Higher Algebra|Higher Algebra, remark 6.2.0.7, prop. 6.2.2.3]], \hyperlink{RiehlVerity13}{Riehl-Verity 13, page 6}). (Of course there is, instead, extra data/information carried by the choice of $\mathcal{D}$.) This should justify the [[simplicial model category]]-theoretic discussion in (\hyperlink{Hess10}{Hess 10}) in [[(∞,1)-category theory]].

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

\begin{itemize}%
\item [[higher monadic descent]]


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


\item [[monad]] / [[2-monad]]/ [[doctrine]] / \textbf{$(\infty,1)$-monad}

\begin{itemize}%
\item [[idempotent (∞,1)-monad]]


\item [[modal type theory]]



\end{itemize}

\item [[operad]] / [[(∞,1)-operad]]



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

A general treatment of $(\infty,1)$-monads in [[(∞,1)-category theory]] is in

\begin{itemize}%
\item [[Jacob Lurie]], section 3 of \emph{Noncommutative algebra} (\href{http://arxiv.org/abs/math/0702299}{math.CT/0702299})

\end{itemize}
later absorbed as

\begin{itemize}%
\item [[Jacob Lurie]], section 6.2 of \emph{[[Higher Algebra]]}

\end{itemize}
More explict discussion in terms of [[quasi-categories]] and [[simplicial sets]] is in

\begin{itemize}%
\item [[Emily Riehl]], [[Dominic Verity]], \emph{Homotopy coherent adjunctions and the formal theory of monads} (\href{http://arxiv.org/abs/1310.8279}{arXiv:1310.8279})

\end{itemize}
Some homotopy theory of ([[enriched functor|enriched]]) monads on ([[simplicial model category|simplicial]]) [[model categories]] is discussed (with an eye towards [[higher monadic descent]]) in

\begin{itemize}%
\item [[Kathryn Hess]], \emph{A general framework for homotopic descent and codescent}, \href{http://arxiv.org/abs/1001.1556}{arXiv/1001.1556}

\end{itemize}
[[!redirects (∞,1)-monad]]

[[!redirects (infinity,1)-monads]] [[!redirects (∞,1)-monads]]

[[!redirects (∞,1)-comonad]] [[!redirects (∞,1)-comonads]]

[[!redirects (infinity,1)-comonad]] [[!redirects (infinity,1)-comonads]]



\end{document}