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\section*{graded monad}

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{uses}{Uses}\dotfill \pageref*{uses} \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}

By analogy with [[graded algebra]], a \textbf{$\mathcal{M}$-graded monad} in a category $\mathcal{C}$ for a [[monoidal category]], $(\mathcal{M}, \otimes, I)$, is a [[lax monoidal functor]], $(\mathcal{M}, \otimes, I) \to ([\mathcal{C}, \mathcal{C}], \circ, id_{\mathcal{C}})$. This generalizes the concept of [[monad]] which may be consider as graded by $\mathbf{1}$, the [[terminal category]]. This definition may be rephrased in terms of a lax action of $\mathcal{M}$ on $\mathcal{C}$.

Equivalently, a $\mathcal{M}$-graded monad is a lax [[2-functor]] from the \href{https://ncatlab.org/nlab/show/delooping#deloopings_of_higher_categorical_structures}{delooping} (or ``suspension'') of $\mathcal{M}$, $\mathbf{B} \mathcal{M} \to Cat$. Just as [[monads]] may be defined in any 2-category, $K$, this suggests that we may generalize graded monads to lax 2-functors $\mathbf{B} \mathcal{M} \to K$.

Graded monads are also known as \textbf{parametric} monads. The grading idea may also be applied to [[comonads]].

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

\begin{enumerate}%
\item The grading may arise from a monoid $(M, \otimes, e)$. Then for some given category, $C$, we have a family of endofunctors, $T_m$, indexed by elements of $M$, with maps $\mu_{m, n, X}: T_m(T_n X) \to T_{m \otimes n} X$ and $\eta_{X}: X \to T_{e} X$, for $m, n$ in $M$ and $X$ in $C$. For instance, there is a $(\mathbb{N}, \times, 1)$-graded monad on sets where $T_n$ returns lists of length $n$ of elements of a set.


\item In computer science, monads model [[side effects|effects]] and comonads coeffects. Grading can therefore allow further annotation of these. For instance, there are graded versions of the [[reader monad]], [[state monad]] and [[writer comonad]] (\hyperlink{OrchPet}{OrchPet}).


\item Any [[graded modality]], such as found in [[bounded linear logic]].


\item For graded monads relevant for probability theory see (\hyperlink{Perrone}{Perrone}).


\item Given the strict action of a monoidal category, $\mathcal{M}$ on a category $\mathcal{B}$, and an adjunction



\end{enumerate}
$\backslash$begin\{center\} $\backslash$begin\{tikzcd\} $\backslash$mathcal\{A\} $\backslash$arrowr, shift right=6pt, ``R''', ``$\backslash$bot'' \& $\backslash$mathcal\{B\}, $\backslash$arrowl, shift right=6pt, ``L''' $\backslash$end\{tikzcd\} $\backslash$end\{center\} then $\mathcal{A}$ inherits a lax action of $\mathcal{M}$ and is hence a graded monad. Every lax action can be generated from a strict action in this way. Initial and terminal such resolutions of a lax action then generalize the $\mathcal{M} \cong 1$ situation in which is a monad is resolved into adjunctions with the [[Kleisli category|Kleisli]] and [[Eilenberg-Moore category|Eilenberg-Moore]] categories (\hyperlink{FKM}{FKM 16}).

\hypertarget{uses}{}\subsection*{{Uses}}\label{uses}

Graded monads can be used to construct ordinary monads by left Kan extension in the 2-category of [[monoidal category|monoidal categories]], [[lax monoidal functor|lax monoidal functors]], and [[monoidal natural transformation|monoidal transformations]]. There are known criteria for when this Kan extension exists; see (\hyperlink{FPKan}{Fritz \& Perrone 18, Theorem 2.1}), as well as the references in there on algebraic Kan extensions.

For example, taking the left Kan extension of the graded list monad $\mathbf{B} \mathbb{N} \to [Set,Set]$ described above results in the usual list monad on $Set$, given by a lax monoidal functor $1 \to [Set,Set]$. Based on a similar construction on the category of complete metric spaces, (\hyperlink{FPKant}{Fritz \& Perrone 17}) have contructed a monad of Radon probability measures without any appeal to measure theory; the intuitive idea being that a probability measure can be thought of as an idealized version of a finite sample, and spaces of finite samples make up a graded monad. Forgetting the grading by taking the above Kan extension then produces the \emph{Kantorovich monad}, containing all Radon probability measures of finite first moment. This construction reduces certain problems in measure-theoretic probability to purely combinatorial problems.

A \emph{useful feature} of such constructions is that the multiplication of the graded monad is often a \emph{strong} monoidal functor in practice. For the graded list monad, this is because a list of length $m n$ can be decomposed uniquely into a list of length $m$ of lists of length $n$, so that the multiplication $T_m T_n X \longrightarrow T_{m n} X$ is an isomorphism. For the probability monad mentioned in the previous paragraph, the analogous phenomenon occurs as well, and this can be exploited e.g. in order to prove a disintegration theorem for finite first moment Radon probability measures on complete metric spaces (\hyperlink{Perrone}{Perrone, Theorem 2.6.9}).

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

\begin{itemize}%
\item [[graded modality]]
\item [[monad]]
\item [[actegory]]

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

\begin{itemize}%
\item Soichiro Fujii, Shin-ya Katsumata, and [[Paul-André Melliès]], \emph{Towards a Formal Theory of Graded Monads}, (\href{https://www.irif.fr/~mellies/papers/fossacs2016-final-paper.pdf}{pdf})


\item Soichiro Fujii, \emph{A 2-Categorical Study of Graded and Indexed Monads}, (\href{https://arxiv.org/abs/1904.08083}{arXiv:1904.08083})


\item Ulrich Dorsch, Stefan Milius and Lutz Schröder, \emph{Graded Monads and Graded Logics for the Linear Time – Branching Time Spectrum}, (\href{https://arxiv.org/abs/1812.01317}{arXiv:1812.01317})


\item Paolo Perrone, \emph{Categorical Probability and Stochastic Dominance in Metric Spaces}, (\href{http://www.paoloperrone.org/phdthesis.pdf}{thesis})


\item [[Tobias Fritz]] and Paolo Perrone, \emph{A Criterion for Kan Extensions of Lax Monoidal Functors}, (\href{https://arxiv.org/abs/1809.10481}{arXiv:1809.10481}).


\item [[Tobias Fritz]] and Paolo Perrone, \emph{A Probability Monad as the Colimit of Spaces of Finite Samples}, (\href{https://arxiv.org/abs/1712.05363}{arXiv:1712.05363}).


\item [[Dominic Orchard]], Tomas Petricek, \emph{Embedding effect systems in Haskell}, (\href{https://www.doc.ic.ac.uk/~dorchard/publ/haskell14-effects.pdf}{pdf})



\end{itemize}


\end{document}