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\begin{document}

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\section*{strong monad}

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

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

[[!include higher algebra - contents]]

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

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

\hypertarget{strong_monads}{}\section*{{Strong monads}}\label{strong_monads}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{concrete_definition}{Concrete definition}\dotfill \pageref*{concrete_definition} \linebreak
\noindent\hyperlink{moggis_typing_rules_and_parameterized_definition}{Moggi's typing rules and parameterized definition}\dotfill \pageref*{moggis_typing_rules_and_parameterized_definition} \linebreak
\noindent\hyperlink{uniqueness_of_strength_with_enough_points}{Uniqueness of strength with enough points}\dotfill \pageref*{uniqueness_of_strength_with_enough_points} \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{strong monad} over a [[monoidal category]] $V$ is a [[monad]] in the [[bicategory]] of $V$-[[actegory|actegories]].

If $V$ is a [[monoidal closed category]], then a strong monad is the same thing as a $V$-[[enriched monad]].

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

\begin{defn}
\label{}\hypertarget{}{}
For $V$ a [[monoidal category]] a \textbf{strong monad} over $V$ is a [[monad]]

\begin{itemize}%
\item in the $2$-[[2-category|category]] $V\text{-}Act$ of left $V$-[[actegory|actegories]] on [[category|categories]]


\item on the object $V$ itself.



\end{itemize}
\end{defn}
Here we regard $V$ as equipped with the canonical $V$-action on itself.

\hypertarget{details}{}\subsection*{{Details}}\label{details}

If we write $\mathbf{B}V$ for the one-object [[bicategory]] obtained by [[delooping]] $V$ once, we have

\begin{displaymath}
V\text{-}Act \simeq Lax2Funct(\mathbf{B}V, Cat)
  \,,
\end{displaymath}
where on the right we have the $2$-category of [[lax functor|lax 2-functors]] from $\mathbf{B}V$ to [[Cat]], lax [[natural transformations]] of and [[modifications]].

The category $V$ defines a canonical functor $\hat V : \mathbf{B}V \to Cat$.

The strong monad, being a [[monad]] in this [[lax functor]] [[bicategory]] is given by

\begin{itemize}%
\item a lax [[natural transformation]] $T : \hat V \to \hat V$;


\item [[modifications]]

\begin{itemize}%
\item unit: $\eta : Id_V \Rightarrow T$


\item product: $\mu : T \circ T \Rightarrow T$



\end{itemize}

\item satisfying the usual uniticity and associativity constraints.



\end{itemize}
By the general logic of $(2,1)$-[[(n,k)-transformation|transformations]] the components of $T$ are themselves a certain [[functor]].

Then the \href{http://en.wikipedia.org/wiki/Strong_monad}{usual diagrams} that specify a strong monad

\begin{itemize}%
\item unitalness and functoriality of the component functor of $T$;


\item naturalness of unit and product modifications.



\end{itemize}
\hypertarget{concrete_definition}{}\subsection*{{Concrete definition}}\label{concrete_definition}

A more concrete definition is given in:

\begin{itemize}%
\item [[Anders Kock]], \emph{Strong functors and monoidal monads}, Arch. Math. (Basel) 23 (1972), 113--120.

\end{itemize}
and later in

\begin{itemize}%
\item [[Eugenio Moggi]]. Computational Lambda-Calculus and Monads. Proceedings of the Fourth Annual Symposium on Logic in Computer Science. 1989. p. 14--23.

\end{itemize}
A \emph{strong monad} over a category $C$ with finite products is a monad $(T, \eta, \mu)$ together with a natural transformation $t_{A,B}$ from $A \times T\;B$ to $T(A\times B)$ subject to three diagrams.

``Strengthening with 1 is irrelevant'':

\begin{displaymath}
\begin{array}{ccc}
1\times T\, A & \rightarrow & T\, A\\
\\
 & t_{1,A}\searrow\phantom{t_{1,A}} & \downarrow\\
\\
 &  & T(1\times A)
\end{array}
\end{displaymath}
``Consecutive applications of strength commute'':

\begin{displaymath}
\begin{array}{ccccc}
(A\times B)\times T\, C & \xrightarrow{t_{A\times B,C}} & T\,((A\times B)\times C)\\
\\
\cong\downarrow\phantom{\cong} &  &  & \phantom{\cong}\searrow\cong\\
\\
A\times(B\times T\, C) & \xrightarrow[A\times t_{B,C}]{} & A\times T\,(B\times C) & \xrightarrow[t_{A,B\times C}]{} & T(A\times(B\times C))
\end{array}
\end{displaymath}
``Strength commutes with monad unit and multiplication'':

\begin{displaymath}
\begin{array}{ccccc}
A\times B\\
\\
A\times\eta_{B}\downarrow\phantom{A\times\eta_{B}} & \phantom{\eta_{A\times B}}\searrow\eta_{A\times B}\\
\\
A\times T\, B & \xrightarrow{t_{A,B}} & T(A\times B)\\
\\
A\times\mu_{B}\uparrow\phantom{A\times\mu_{B}} &  &  & \phantom{\mu_{A\times B}}\nwarrow\mu_{A\times B}\\
\\
A\times T^{2}\, B & \xrightarrow{t_{A,T B}} & T(A\times T B) & \xrightarrow{T\: t_{A,B}} & T^{2}(A\times B)
\end{array}
\end{displaymath}
More generally, if a monoidal category $V$ acts on a category $C$

\begin{displaymath}
\bullet : V\times C\to C
\end{displaymath}
then a $V$-strength for a monad $T$ on $C$ is a family of morphisms $t_{A,B}:A\bullet T(B)\to T(A\bullet B)$ satisfying similar commutative diagrams.

\hypertarget{moggis_typing_rules_and_parameterized_definition}{}\subsection*{{Moggi's typing rules and parameterized definition}}\label{moggis_typing_rules_and_parameterized_definition}

Moggi proposed the following typing rules for a sequence operator:

\begin{displaymath}
\frac{\Gamma \vdash t:X}{\Gamma\vdash \eta(t):T(X)}
\qquad
\frac{\Gamma,x:X \vdash u: T(Y)\quad \Delta\vdash t:T(X)}
{\Gamma,\Delta\vdash let\,x=t\,in\,u:T(Y)}
\end{displaymath}
To interpret these rules in a category, where types are interpreted as objects and judgements are interpreted as morphisms, we require

\begin{itemize}%
\item For each object $X$, an object $T(X)$;


\item a morphism $\eta_X:X\to T(X)$ for each $X$ (equivalently, a natural family of functions $C(\Gamma,X)\to C(\Gamma, T(X))$);


\item a family of functions $*_{\Gamma,\Delta}:C(\Gamma\otimes X, T(Y))\times C(\Delta,T(X))\to C(\Gamma\otimes \Delta,T(Y))$ natural in $\Gamma$ and $\Delta$,



\end{itemize}
such that

\begin{itemize}%
\item $f * \eta =f$, $\eta * f=f$, and $h*(g*f)=(h*g)*f$.

\end{itemize}
With the subscripts:

\begin{itemize}%
\item $f *_{\Gamma,X} \eta_X = f$ and $\eta *_{I,\Delta} f=f$, and $h *_{\Gamma,\Delta\otimes \Xi} (g *_{\Delta,\Xi} f) = (h *_{\Gamma,\Delta\otimes X} g) *_{\Gamma\otimes\Delta,\Xi} f$.

\end{itemize}
Such a structure is the same thing as a strong monad. One way to see this is to notice that it is essentially the same as the Kleisli triple form of an $\hat C$-[[enriched monad]] on $C$, where $\hat C$ is the category of [[presheaves]] on $C$ regarded with the [[Day convolution]] monoidal structure. More concretely,

\begin{displaymath}
\eta_{X\otimes Y}*_{X,T(Y)} id_{T(Y)}:X\otimes T(Y)\to T(X\otimes Y)
\end{displaymath}
is a strength map.

\hypertarget{uniqueness_of_strength_with_enough_points}{}\subsection*{{Uniqueness of strength with enough points}}\label{uniqueness_of_strength_with_enough_points}

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(Moggi)}

Let $C$ be a category with finite products and let $T$ be a strong monad on $C$. For any points $x:1\to X$, $y:1\to T(Y)$, we have

\begin{displaymath}
t\circ(x,y)\ = \ 
T((x\circ !_Y),id_Y)\circ y\ : 1 \to T(X\times Y)
\end{displaymath}
Hence if $1$ is a [[generator]], i.e. $C(1,-):C\to Set$ is faithful, then there is at most one strength for any ordinary monad on $C$.

\end{theorem}
In other words, a monad being strong is a property rather than structure in a category with enough points.

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

\begin{itemize}%
\item [[tensorial strength]]


\item [[monoidal monad]]


\item [[commutative monad]]


\item [[enriched monad]]


\item [[monad (in computer science)]]



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

Strong monads were defined by Kock, as an alternative description of enriched monads.

\begin{itemize}%
\item [[Anders Kock]], \emph{Strong functors and monoidal monads}, Arch. Math. (Basel) 23 (1972), 113--120.

\end{itemize}
Usually strong monads are described explicitly in terms of the components of the above structure. The above repackaging of that definition appears in the blog post

\begin{itemize}%
\item [[John Baez]], \emph{The Monads Hurt My Head -- But Not Anymore} (\href{http://golem.ph.utexas.edu/category/2009/07/the_monads_hurt_my_head_but_no.html#c025476}{blog})

\end{itemize}
Strong monads are important in Moggi's theory of notions of computation (see [[monad (in computer science)]]):

\begin{itemize}%
\item [[Eugenio Moggi]]. Notions Of Computation And Monads. Information And Computation. 1991;93:55--92.
\item [[Eugenio Moggi]]. Computational Lambda-Calculus and Monads. Proceedings of the Fourth Annual Symposium on Logic in Computer Science. 1989. p. 14--23.

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



\end{document}