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\begin{document}

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\section*{monoidal monad}

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

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

[[!include higher algebra - contents]]

\hypertarget{monoidal_categories}{}\paragraph*{{Monoidal categories}}\label{monoidal_categories}

[[!include monoidal categories - contents]]

\hypertarget{2category_theory}{}\paragraph*{{2-Category theory}}\label{2category_theory}

[[!include 2-category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{tensorial_strengths_and_commutative_monads}{Tensorial strengths and commutative monads}\dotfill \pageref*{tensorial_strengths_and_commutative_monads} \linebreak
\noindent\hyperlink{from_monoidal_monads_to_commutative_monads}{From monoidal monads to commutative monads}\dotfill \pageref*{from_monoidal_monads_to_commutative_monads} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{monoidal monad} is a [[monad]] in the [[2-category]] of [[monoidal category|monoidal categories]], [[lax monoidal functor|lax monoidal functors]], and monoidal transformations.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The notion of monoidal monad is equivalent to a suitable general notion of \emph{[[commutative monad]]} (see def. \ref{CommutativeMonad} below), as discussed at [[commutative algebraic theory]]. We explore this connection below.

\end{remark}
\hypertarget{tensorial_strengths_and_commutative_monads}{}\subsection*{{Tensorial strengths and commutative monads}}\label{tensorial_strengths_and_commutative_monads}

As a preliminary, let $V$ be a [[monoidal category]]. We say a functor $T \colon V \to V$ is \textbf{strong} if there are given left and right [[tensorial strength|tensorial strengths]]

\begin{displaymath}
\tau_{A, B} \colon A \otimes T(B) \to T(A \otimes B)
\end{displaymath}
\begin{displaymath}
\,
\end{displaymath}
\begin{displaymath}
\sigma_{A, B} \colon T(A) \otimes B \to T(A \otimes B).
\end{displaymath}
which are suitably compatible with one another. The full set of [[coherence]] conditions may be summarized by saying $T$ preserves the two-sided monoidal [[action]] of $V$ on itself, in an appropriate [[2-category theory|2-categorical]] sense. More precisely: the two-sided action of $V$ on itself is a [[lax functor]] of 2-categories

\begin{displaymath}
\tilde{V} \colon B V \times (B V)^{op} \to Cat
\end{displaymath}
($B V$ is the one-object 2-category associated with a monoidal category $V$, and $(B V)^{op}$ is the same 2-category but with 1-cell composition (= tensoring) in reverse order), and the two-sided strength means we have a structure of [[lax natural transformation]] $\tilde{V} \to \tilde{V}$.

\begin{remark}
\label{}\hypertarget{}{}
In the setting where $V$ is \emph{[[symmetric monoidal category|symmetric]]} monoidal, we will assume that the left and right strengths $\tau$ and $\sigma$ are related by the symmetry in the obvious way, by a commutative square

\begin{displaymath}
\itexarray{
A \otimes T(B) & \stackrel{\tau_{A, B}}{\to} & T(A \otimes B) \\
 ^\mathllap{c} \downarrow & & \downarrow^\mathrlap{T(c)} \\
T(B) \otimes A & \underset{\sigma_{B, A}}{\to} & T(B \otimes A)
}
\end{displaymath}
where the $c$`s are instances of the symmetry isomorphism.

\end{remark}
There is a category of strong functors $V \to V$, where the morphisms are transformations $\lambda \colon S \to T$ which are compatible with the strengths in the obvious sense. Under composition, this is a strict monoidal category.

\begin{defn}
\label{StrongMonad}\hypertarget{StrongMonad}{}
Monoids in this monoidal category are called \textbf{[[strong monads]]}.

\end{defn}
\begin{defn}
\label{CommutativeMonad}\hypertarget{CommutativeMonad}{}
A [[strong monad]] $(T \colon V \to V, m \colon T T \to T, u: 1 \to T)$ (def. \ref{StrongMonad}) is a \textbf{[[commutative monad]]} if there is an equality of [[natural transformations]] $\alpha = \beta$ where

\begin{itemize}%
\item $\alpha$ is the composite

\begin{displaymath}
T A \otimes T B \stackrel{\sigma_{A, T B}}{\to} T(A \otimes T B) \stackrel{T(\tau_{A, B})}{\to} T T(A \otimes B) \stackrel{m(A \otimes B)}{\to} T(A \otimes B).
\end{displaymath}

\item $\beta$ is the composite

\begin{displaymath}
T A \otimes T B \stackrel{\tau_{T A, B}}{\to} T(T A \otimes B) \stackrel{T(\sigma_{A, B})}{\to} T T(A \otimes B) \stackrel{m(A \otimes B)}{\to} T(A \otimes B).
\end{displaymath}


\end{itemize}
\end{defn}
\hypertarget{from_monoidal_monads_to_commutative_monads}{}\subsection*{{From monoidal monads to commutative monads}}\label{from_monoidal_monads_to_commutative_monads}

Let $(T \colon V \to V, u \colon 1 \to T, m \colon T T \to T)$ be a monoidal monad, with structural constraints on the underlying functor denoted by

\begin{displaymath}
\alpha_{A, B} \colon T(A) \otimes T(B) \to T(A \otimes B), \qquad \iota = u I: I \to T(I).
\end{displaymath}
Define [[tensorial strength|strengths]] on both the left and the right by

\begin{displaymath}
\tau_{A, B} = (A \otimes T(B) \stackrel{u A \otimes 1}{\to} T(A) \otimes T(B) \stackrel{\alpha_{A, B}}{\to} T(A \otimes B)),
\end{displaymath}
\begin{displaymath}
\,
\end{displaymath}
\begin{displaymath}
\sigma_{A, B} = (T(A) \otimes B \stackrel{1 \otimes u B}{\to} T(A) \otimes T(B) \stackrel{\alpha_{A, B}}{\to} T(A \otimes B)).
\end{displaymath}
\begin{prop}
\label{}\hypertarget{}{}
$(m \colon T T \to T, u \colon 1 \to T)$ is a commutative monad.

\end{prop}
\begin{proof}
In fact, the two composites

\begin{displaymath}
T A \otimes T B \stackrel{\sigma_{A, T B}}{\to} T(A \otimes T B) \stackrel{T(\tau_{A, B})}{\to} T T(A \otimes B) \stackrel{m(A \otimes B)}{\to} T(A \otimes B)
\end{displaymath}
\begin{displaymath}
\,
\end{displaymath}
\begin{displaymath}
T A \otimes T B \stackrel{\tau_{T A, B}}{\to} T(T A \otimes B) \stackrel{T(\sigma_{A, B})}{\to} T T(A \otimes B) \stackrel{m(A \otimes B)}{\to} T(A \otimes B)
\end{displaymath}
are both equal to $\alpha_{A, B}$. We show this for the first composite; the proof is similar for the second. If $\alpha_T$ denotes the monoidal constraint for $T$ and $\alpha_{T T}$ the constraint for the composite $T T$, then by definition $\alpha_{T T}$ is the composite given by

\begin{displaymath}
T T X \otimes T T Y \stackrel{\alpha_T T}{\to} T(T X \otimes T Y) \stackrel{T\alpha_T}{\to} T T(X \otimes Y)
\end{displaymath}
and so, using the properties of monoidal monads, we have a commutative diagram

\begin{displaymath}
\itexarray{
 & & T T X \otimes T Y & \stackrel{\alpha_T}{\to} & T(T X \otimes Y) \\
 & ^\mathllap{u \otimes 1} \nearrow & \downarrow^\mathrlap{1 \otimes T u} & & \downarrow^\mathrlap{T(1 \otimes u)} \\
T X \otimes T Y & \stackrel{u \otimes T u}{\to} & T T X \otimes T T Y & \stackrel{\alpha_T T}{\to} & T(T X \otimes T Y) \\
 & ^\mathllap{1} \searrow & \downarrow^\mathrlap{m \otimes m} & \searrow^\mathrlap{\alpha_{T T}} & \downarrow^\mathrlap{T\alpha_T} \\
 & & T X \otimes T Y & & T T(X \otimes Y) \\
& & & ^\mathllap{\alpha_T} \searrow & \downarrow^\mathrlap{m} \\
 & & & & T(X \otimes Y)
}
\end{displaymath}
which completes the proof.

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

\begin{itemize}%
\item Gavin J. Seal, \emph{Tensors, monads and actions} (\href{http://arxiv.org/abs/1205.0101}{arXiv:1205.0101})

\end{itemize}
[[!redirects monoidal monads]]



\end{document}