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\section*{Kantorovich monad}

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

\hypertarget{measure_and_probability_theory}{}\paragraph*{{Measure and probability theory}}\label{measure_and_probability_theory}

[[!include measure theory - contents]]

\hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

\hypertarget{category_theory}{}\paragraph*{{$(0,1)$-Category theory}}\label{category_theory}

[[!include (0,1)-category theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{wasserstein_spaces}{Wasserstein spaces}\dotfill \pageref*{wasserstein_spaces} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{functor_unit_and_multiplication}{Functor, unit and multiplication}\dotfill \pageref*{functor_unit_and_multiplication} \linebreak
\noindent\hyperlink{algebras}{Algebras}\dotfill \pageref*{algebras} \linebreak
\noindent\hyperlink{duality}{Duality}\dotfill \pageref*{duality} \linebreak
\noindent\hyperlink{the_ordered_case}{The ordered case}\dotfill \pageref*{the_ordered_case} \linebreak
\noindent\hyperlink{for_lawvere_metric_spaces}{For Lawvere metric spaces}\dotfill \pageref*{for_lawvere_metric_spaces} \linebreak
\noindent\hyperlink{see_also}{See also}\dotfill \pageref*{see_also} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The Kantorovich monad is a [[probability monad]] on the category of [[complete metric spaces]] and 1-Lipschitz (or [[Lipschitz map|Lipschitz]]) maps. Its functor part assigns to each complete metric space $X$ the space of [[Radon measure|Radon]] [[probability measures]] with finite first moment, equipped with the [[Kantorovich-Wasserstein distance]].

Its [[algebra over a monad|algebras]] are [[closed set|closed]] [[convex set|convex]] subsets of [[Banach spaces]].

The construction was first given by van Breugel for the compact case (see \hyperlink{vanbreugel}{vB `05}).

There is an [[order theory|ordered]] version of the Kantorovich monad, making use of the [[stochastic order]]. Its [[algebra over a monad|algebras]] are [[closed set|closed]] [[convex set|convex]] subsets of [[ordered Banach spaces]].

\hypertarget{wasserstein_spaces}{}\subsection*{{Wasserstein spaces}}\label{wasserstein_spaces}

Let $X$ be a [[metric space]]. A [[probability measure]] on $X$ has \textbf{finite first moment} if

\begin{displaymath}
\int_{X\times X} d(x,y) \, dp(x)\,dp(y) \;\lt\; +\infty,
\end{displaymath}
where $d(-,-)$ denotes the [[distance]].

Let $p$ and $q$ be [[Radon measure|Radon]] [[probability measures]] on $X$ of finite first moment. The \textbf{[[Kantorovich-Wasserstein distance]]} is given by

\begin{displaymath}
d(p,q) \;\coloneqq\; \inf_{r\in\Gamma(p,q)} \int_{X\times X} d(x,y) \,dr(x,y) ,
\end{displaymath}
where the infimum (which is attained if $X$ is complete) is taken over the set $\Gamma(p,q)$ of [[joint measure|couplings]] of $p$ and $q$, i.e. the measures on $X\times X$ which admit $p$ and $q$ as their [[marginal measure|marginals]].

Equivalently, the distance can be written as

\begin{displaymath}
d(p,q) \;=\; \sup_{f\in[X,\mathbb{R}]} \int f\, dp - \int f\, dq,
\end{displaymath}
where the supremum (which is usually not attained) is taken over the set of 1-[[Lipschitz map|Lipschitz]] functions $X\to \mathbb{R}$. (This equivalence is an instance of the celebrated [[Kantorovich duality]].)

The \textbf{1-Wasserstein space} over $X$ is the space $P X$ of Radon probability measures on $X$ with finite first moment, equipped with the Wasserstein metric.

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

\begin{itemize}%
\item If $X$ is [[complete metric space|complete]], so is $P X$.


\item If $X$ is [[separable space|separable]], so if $P X$.


\item If $X$ is [[compact]], so is $P X$.



\end{itemize}
More information on Wasserstein spaces can be found in \hyperlink{villani}{Villani `08}.

\hypertarget{functor_unit_and_multiplication}{}\subsection*{{Functor, unit and multiplication}}\label{functor_unit_and_multiplication}

As it is customary for [[probability monads]], the functor assigns to a [[morphism]] $f:X\to Y$ (in this case, a [[Lipschitz map|Lipschitz]] or 1-Lipschitz map) the map $P f: P X \to P Y$ induced by the [[pushforward of measures]]. The pushforward of a Radon probability measure of finite first moment along a Lipschitz map is again Radon and of finite first moment. Moreover, the resulting map $P f$ is again Lipschitz, with the same Lipschitz constant as $f$. Therefore we have an endofunctor $P$ on the category of [[complete metric spaces]] and [[Lipschitz maps]] (or 1-Lipschitz).

Again, as it happens in general with [[probability monads]], the unit $\delta:X\to P X$ is given by forming the [[Dirac measures]] and the multiplication $E:P P X\to P X$ is given by [[integration]]. More formally, given $\xi\in P P X$, the measure $E \xi\in PX$ is defined as the one which assigns to a [[Borel set]] $A\subseteq X$ the number

\begin{displaymath}
E \xi(A) \;\coloneqq\; \int p(A) \,d\xi(p) .
\end{displaymath}
The maps $\delta$ and $E$ satisfy the usual [[axioms]] of a [[monad]] (see \hyperlink{kantorovich19}{F-P `19}), which is known as the \textbf{Kantorovich monad}.

\hypertarget{algebras}{}\subsection*{{Algebras}}\label{algebras}

As it is the case for most [[probability monad]], the algebras are [[convex spaces]] of a particular kind. For the Kantorovich monad, the objects of the underlying category are complete metric spaces, and so the [[closed set|closed]] [[convex set|convex]] subsets of [[Banach spaces]] are an ideal candidate: they are [[metric spaces]], they are [[complete metric space|complete]], and they have a well-defined [[convex space|convex combination operation]]. It turns out that this is exactly the case.

More in detail, every closed convex subset $A$ of a Banach space comes equipped with a map $e:P A\to A$ given by vector-valued [[integration]],

\begin{displaymath}
p\;\mapsto\; \int a \,dp(a) .
\end{displaymath}
The integral exists since $p$ has finite first moment. Conversely, it can be proven that every $P$-algebra is of this form.

The [[Eilenberg-Moore category|morphisms of algebras]] are maps between algebras $f:A\to B$ which commute with the integration map. It can be shown that equivalently these are exactly the \textbf{affine} maps, i.e.{\tt \symbol{126}}those maps which satisfy

\begin{displaymath}
f\big(\lambda\,x + (1-\lambda)\,y\big) \;=\; \lambda\,f(x) + (1-\lambda)\,f(y).
\end{displaymath}
For more details, see \hyperlink{kantorovich19}{F-P `19}.

\hypertarget{duality}{}\subsection*{{Duality}}\label{duality}

(\ldots{})

\hypertarget{the_ordered_case}{}\subsection*{{The ordered case}}\label{the_ordered_case}

(Work in progress. For now see \hyperlink{orderedkantorovich}{F-P `18}.)

\hypertarget{for_lawvere_metric_spaces}{}\subsection*{{For Lawvere metric spaces}}\label{for_lawvere_metric_spaces}

(\ldots{})

\hypertarget{see_also}{}\subsection*{{See also}}\label{see_also}

\begin{itemize}%
\item [[monads of probability, measures, and valuations]]


\item [[Giry monad]], [[Radon monad]], [[extended probabilistic powerdomain]]


\item [[Kantorovich-Wasserstein distance]], [[optimal transport]]


\item [[metric space]], [[complete metric space]]


\item [[Banach space]], [[ordered Banach space]], [[convex space]]



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

\begin{itemize}%
\item Franck van Breugel, \emph{The metric monad for probabilistic nondeterminism}, unpublished, 2005. (\href{http://www.cse.yorku.ca/~franck/research/drafts/monad.pdf}{pdf})


\item [[Tobias Fritz]] and Paolo Perrone, \emph{A probability monad as the colimit of spaces of finite samples}, Theory and Applications of Categories 34, 2017. (\href{http://www.tac.mta.ca/tac/volumes/34/7/34-07.pdf}{pdf})


\item [[Tobias Fritz]] and Paolo Perrone, \emph{Stochastic order on metric spaces and the ordered Kantorovich monad}, submitted, 2018. (\href{https://arxiv.org/abs/1808.09898}{arXiv:1808.09898})


\item [[Cedric Villani]], \emph{Optimal transport: old and new}, Springer, 2008.


\item Paolo Baldan, Filippo Bonchi, Henning Kerstan and Barbara König, \emph{Coalgebraic behavioral metrics}, Logical Methods in Computer Science 14(3), 2018. (\href{https:doi.org/10.23638/LMCS-14%283:20%292018}{doi: 10.23638/LMCS-14(3:20)2018})



\end{itemize}


\end{document}