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\begin{document}

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\section*{Monads and the Barr-Beck Theorem}

\hypertarget{31_categories_of_endofunctors}{}\subsection*{{3.1 $(\infty,1)$-Categories of Endofunctors}}\label{31_categories_of_endofunctors}

For an $(\infty,1)$-category $M$ the $(\infty,1)$-category $Fun(M,M)$ is a monoid object in the category $sSet$ and $M$ is endowed with a ($1$-categorial) left action of $Fun(M,M)$ and this action is universal among left actions on $M$.

Recall how the $(\infty,1)$-Grothendieck construction works in the following example: Let $I$ be a $1$-category, let $f:N(I)\to (\infty,1)Cat$ be a diagram. We obtain the desired cartesian fibration $X\to N(I)$ by first replacing $f$ by a simplicial functor $F:N_{coh}(I)^{op}\to sSet^+$ where $N_{coh}$ denotes the [[nLab:homotopy coherent nerve]] functor and $sSet^+$ denotes the category of [[nLab:marked simplicial set]]s. $F$ is a weakly fibrant object of $(sSet^+)^{N_{coh}(I)^{op}}$. Applying the [[nLab:unstraightening functor]] $Un^+_{N(J)}$ we obtain a fibrant object $sSet^+/N(I)$ which we identify with the desired cartesian fibration $p:X\to N(I)$.

This statement shall be lifted to $(\infty,1)$.

\hypertarget{preparation}{}\subsection*{{Preparation}}\label{preparation}

\begin{udefn}
Let $C$ be a symmetric monoidal category with tensor $\otimes$. The category $C^\otimes$ consists of the following data:

(1) Objects are finite sequences of $C$-objects $[C_1,\dots,C_n]$.

(2) Morphisms $f:[C_1,\dots,C_n]\to [C_1^',\dots C_m^']$ are pairs

\begin{displaymath}
(a_S:S\to\{1,\dots,m\}, (f_j:\otimes_{a(i)=j}C_i\to C_j^')_{1\le j\le m})
\end{displaymath}
where $S\subseteq \{1,\dots,n\}$ is a subset (or rather isomorphic in $Set$ to a subset) .

(3) Composition of $f:=(a_S,(f_i)_i):[C_1,\dots,C_n]\to [C_1^',\dots C_m^']$ and $g:=(b_T,(g_j)_j):[C_1^',\dots,C_m^']\to [C_1^{''},\dots C_l^{''}]$ is defined to be

\begin{displaymath}
g\circ f:=(c_{a^{-1}(T)},\otimes_{b\circ a)(i)=k}C_i\simeq \otimes_{b(j)=k}\otimes_{a(i)=j} C_i\to \otimes_{b(j)=k}C_j^'\to C_k^{''})
\end{displaymath}
for $1\le k\le l$.

A \emph{monoidal $(\infty,1)$-category} is defined to be a cocartesian fibration $p:C^\otimes\to N(\Delta)^{op}$ such that:

\begin{itemize}%
\item For all $n\ge 0$, the associated functors $C^\otimes_{[n]}\to C^\otimes_{\{i,i+1\}}$ determine an equivalence of $(\infty,1)$-categories\begin{displaymath}
C^\otimes_{[n]}\to C^\otimes_{\{0,1\}}\times \dots\times C^\otimes_{\{n-1,n\}}\simeq (C^\otimes_{[1]})^n.
\end{displaymath}


\end{itemize}
where $C^\otimes_{[n]}:=p^{-1}([n)$ denotes the fiber of the forgetful functor $p:[C_1,\dots,C_n]\mapsto [n]$ over $[n]$.

\end{udefn}
In particular for $S:=\{C,D\}$ a set with two distinct we obtain:

\begin{udefn}
\end{udefn}
\hypertarget{definition_2}{}\subsection*{{Definition}}\label{definition_2}

\begin{udefn}
Let $I$ be a category, let $f:I\to sSet$ be a functor. The \emph{nerve of $I$ relative $f$} denoted by $N_f(I)$ is defined as follows: Let $J$ be a finite linear order, the a map $\Delta^J\to N_f(I)$ consists of:

\begin{enumerate}%
\item a functor $s:J\to I$


\item for every nonempty subset $J^\prime\subset J$ having a maximal element $j^\prime$, a map $\tau(J^\prime):\Delta^{J}\to f(\sigma(j^\prime))$.


\item satisfying properties.



\end{enumerate}
\end{udefn}
[[mapping simplex]]: Let $\phi:A^0\leftarrow A^1\leftarrow \dots\leftarrow A^n$ be a composable sequence of maps of simplicial sets. The \emph{mapping simplex of $\phi$} is denoted by $M(\phi)$.

\begin{udefn}
Let $M$ be a simplicial set. Let $End^{\otimes}(M):=N_E(\Delta^{op})$ and $\overline{End^\otimes}(M):=N_{\overline E}(\Delta^{op})$.

Let now $M$ be a $(\infty,1)$-category.

\begin{enumerate}%
\item The map $p:End^{\otimes}(M)\to N(\Delta)^{op}$ determines a monoidal structure on the $(\infty,1)$-category $Fun(M,M)\simeq End^\otimes_{[1]}(M)$.


\item The map $q:\overline{End^\otimes}\to End^\otimes(M)$ exhibits $M\simeq \overline{End^\otimes_{[0]}}(M)$ as left tensored over $Fun(M,M)$.



\end{enumerate}
This monoidal structure on $Fun(M,M)$ is called the \emph{composition monoidal structure}.

\end{udefn}
\begin{udefn}
Let $M$ be an $(\infty,1)$-category. Then a \emph{monad on $M$} is defined to an algebra object in $Fun(M,M)$

\end{udefn}


\end{document}