Add section on functors and natural transformations
Also do not use ugly overbar
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@ -1261,6 +1261,55 @@ That in any category:
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\prod_{A\ B \tp \Object} \isProp\ (\var{Product}\ \bC\ A\ B)
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\end{align}
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%
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\section{Functors and natural transformations}
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For the sake of completeness I will mention the definition of functors
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and natural transformations. Please refer to the implementation for
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the full details.
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%
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\subsection{Functors}
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Given two categories $\bC$ and $\bD$ a functor consists of the
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following data:
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%
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\begin{align*}
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\omapF & \tp ℂ.\Object → 𝔻.\Object \\
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\fmap & \tp ℂ.\Arrow\ A\ B → 𝔻.\Arrow\ (\omapF\ A)\ (\omapF\ B)
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\end{align*}
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%
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And the following laws:
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\begin{align*}
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\fmap\ ℂ.\identity & ≡ 𝔻.identity \\
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\fmap\ (g \clll f) & ≡ \fmap\ g \dlll \fmap\ f
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\end{align*}
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%
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The implementation can be found here:
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%
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\begin{center}
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\sourcelink{Cat.Category.Functor}
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\end{center}
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\subsection{Natural Transformation}
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Given two functors between categories $\bC$ and $\bD$. Name them
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$\FunF$ and $\FunG$. A natural transformation is a family of arrows:
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%
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\begin{align*}
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\prod_{C \tp ℂ.\Object} \bD.\Arrow\ (\omapF\ C)\ (\omapG\ C)
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\end{align*}
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%
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This family of arrows can be seen as the data. If $\theta$ is a
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natural transformation $\theta\ C$ will be called the component (of
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$\theta$) at $C$. The laws of this family of morphism is the
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naturality condition:
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%
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\begin{align*}
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\prod_{f \tp ℂ.\Arrow\ A\ B}
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(θ\ B) \dlll (\FunF.\fmap\ f) ≡ (\FunG.\fmap\ f) \dlll (θ\ A)
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\end{align*}
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%
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The implementation can be found here:
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%
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\begin{center}
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\sourcelink{Cat.Category.NaturalTransformation}
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\end{center}
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\section{Monads}
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\label{sec:monads}
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In this section I present two formulations of monads. The two representations
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@ -1269,8 +1318,14 @@ simply monoidal monads and Kleisli monads for short. We then show that the two
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formulations are equivalent, which due to univalence gives us a path between the
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two types.
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Let a category $\bC$ be given. In the remainder of this sections all objects and
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arrows will implicitly refer to objects and arrows in this category.
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Let a category $\bC$ be given. In the remainder of this sections all
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objects and arrows will implicitly refer to objects and arrows in this
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category. I will also use the notation $\EndoR$ to refer to an
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endofunctor on this category. Its map on objects will be denoted
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$\omapR$ and its map on arrows will be denoted $\fmap$. Likewise I
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will use the notation $\pureNT$ to refer to a natural transformation
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and its component at a given (implicit) object will be denoted
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$\pure$.
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%
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\subsection{Monoidal formulation}
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The monoidal formulation of monads consists of the following data:
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@ -1278,7 +1333,7 @@ The monoidal formulation of monads consists of the following data:
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\begin{align}
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\label{eq:monad-monoidal-data}
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\begin{split}
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\EndoR & \tp \Endo ℂ \\
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\EndoR & \tp \Functor\ ℂ\ \bC \\
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\pureNT & \tp \NT{\EndoR^0}{\EndoR} \\
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\joinNT & \tp \NT{\EndoR^2}{\EndoR}
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\end{split}
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@ -1321,8 +1376,15 @@ The Kleisli-formulation consists of the following data:
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\end{split}
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\end{align}
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%
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The objects $X$ and $Y$ are implicitly universally quantified.
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The objects $X$ and $Y$ are implicitly universally quantified. With this data we can construct the \nomenindex{Kleisli arrow}:
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%
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\begin{align*}
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\fish & \tp \Arrow\ A\ (\omapR\ B)
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\to \Arrow\ B\ (\omapR\ C)
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\to \Arrow\ A\ (\omapR\ C) \\
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f \fish g & \defeq f \rrr (\bind\ g)
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\end{align*}
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%
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It is interesting to note here that this formulation does not talk about natural
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transformations or other such constructs from category theory. All we have here
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is a regular maps on objects and a pair of arrows.
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@ -1339,11 +1401,10 @@ This data must satisfy:
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\end{align}
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\newcommand\kleislilaws{\ref{eq:monad-kleisli-laws-0}, \ref{eq:monad-kleisli-laws-1} and \ref{eq:monad-kleisli-laws-2}}%
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%
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Here likewise the arrows $f \tp \Arrow\ X\ (\EndoR\ Y)$ and $g \tp
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\Arrow\ Y\ (\EndoR\ Z)$ are universally quantified (as well as the objects they
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range over). $\fish$ is the Kleisli-arrow which is defined as $f \fish g \defeq
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f \rrr (\bind\ g)$ . (\TODO{Better way to typeset $\fish$?})
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Here likewise the arrows $f \tp \Arrow\ X\ (\omapR\ Y)$ and $g \tp
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\Arrow\ Y\ (\omapR\ Z)$ are universally quantified as well as the
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objects they range over.
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%
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\subsection{Equivalence of formulations}
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%
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The notation I have chosen here in the report
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@ -1376,27 +1437,27 @@ show that $(\omapR, \pure, \bind)$ is indeed a monad in the Kleisli
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form. In the second part we will show the other direction.
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\subsubsection{Monoidal to Kleisli}
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Let $(\EndoR, \pure, \join)$ be given as in \ref{eq:monad-monoidal-data}
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Let $(\EndoR, \pureNT, \joinNT)$ be given as in \ref{eq:monad-monoidal-data}
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satisfying the laws \monoidallaws. For the data of the Kleisli
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formulation we pick:
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%
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\begin{align}
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\begin{split}
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\EndoR & \defeq \EndoRX \\
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\pure & \defeq \pureX \\
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\bind\ f & \tp \joinX \lll \fmapX\ f
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\omapR & \defeq \omapR \\
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\pure & \defeq \pure \\
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\bind\ f & \defeq \join \lll \fmap\ f
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\end{split}
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\end{align}
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%
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$\EndoRX$ is the object map of the endo-functor $\EndoR$, $\pureX$ and
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$\joinX$ are the arrows from the natural transformations $\pure$ and
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$\join$ respectively. The term $\fmapX$ is the arrow map of the
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endo-functor $\EndoR$. It now just remains to verify the laws
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Again $\omapR$ is the object map of the endo-functor $\EndoR$, $\pure$
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and $\join$ are the arrows from the natural transformations $\pureNT$
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and $\joinNT$ respectively and $\fmap$ is the map on arrows of the
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endofunctor $\EndoR$. It now just remains to verify the laws
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\kleislilaws. For \ref{eq:monad-kleisli-laws-0}:
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%
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\begin{align*}
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\bind\ \pure & ≡
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\join \lll (\fmap\ \pure) && \text{By definition} \\
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\bind\ \pure & =
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\join \lll (\fmap\ \pure) \\
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& ≡ \identity && \text{By \ref{eq:monad-monoidal-laws-2}}
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\end{align*}
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%
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@ -1404,57 +1465,89 @@ For \ref{eq:monad-kleisli-laws-1}:
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%
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\begin{align*}
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\pure \fish f
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& \equiv %%%
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\pure \ggg \bind\ f && \text{By definition} \\ & ≡
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\bind\ f \lll \pure && \text{By definition} \\ & ≡
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\joinX \lll \fmapX\ f \lll \pureX && \text{By definition} \\ & ≡
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\joinX \lll \pureX \lll f && \text{$\pure$ is a natural transformation} \\ & ≡
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& = %%%
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\pure \ggg \bind\ f \\ & =
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\bind\ f \lll \pure \\ & =
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\join \lll \fmap\ f \lll \pure \\ & ≡
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\join \lll \pure \lll f && \text{$\pure$ is a natural transformation} \\ & ≡
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\identity \lll f && \text{By \ref{eq:monad-monoidal-laws-1}} \\ & ≡
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f && \text{Left identity}
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\end{align*}
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%
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For \ref{eq:monad-kleisli-laws-2}:
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\begin{align*}
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\bind\ g \rrr \bind\ f & ≡
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\bind\ g \rrr \bind\ f & =
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\bind\ f \lll \bind\ g
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\\ & ≡
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\\ & =
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%% %%%%
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\joinX \lll \fmapX\ g \lll \joinX \lll \fmapX\ f
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&& \text{\dots} \\ & ≡
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\joinX \lll \joinX \lll \fmapX^2\ g \lll \fmapX\ f
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\join \lll \fmap\ g \lll \join \lll \fmap\ f
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\\ & ≡
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\join \lll \join \lll (\fmap \comp \fmap)\ f \lll \fmap\ g
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&& \text{$\join$ is a natural transformation} \\ & ≡
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\joinX\ \lll \fmapX\ \joinX \lll \fmapX^2\ g \lll \fmapX\ f
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\join \lll \fmap\ \join \lll (\fmap \comp \fmap)\ f \lll \fmap\ g
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&& \text{By \ref{eq:monad-monoidal-laws-0}} \\ & ≡
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\joinX\ \lll \fmapX\ \joinX\ \lll \fmapX\ (\fmapX\ g) \lll \fmapX\ f
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\join \lll \fmap\ \join \lll \fmap\ (\fmap\ f) \lll \fmap\ g
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&& \text{} \\ & ≡
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\joinX \lll \fmapX\ (\joinX \lll \fmapX\ g \lll f)
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&& \text{Distributive law for functors} \\ & \equiv
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\join \lll \fmap\ (\join \lll \fmap\ f \lll g)
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&& \text{Distributive law for functors} \\ & =
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\join \lll \fmap\ (\join \lll \fmap\ f \lll g) \\ & =
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%%%%
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\bind\ (\bind\ f \lll g) \\ & =
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\bind\ (g \rrr \bind\ f) \\ & =
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\bind\ (g \fish f)
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\end{align*}
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%
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The construction can be found in the module:
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\begin{center}
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\sourcelink{Cat.Category.Monad.Monoidal}
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\end{center}
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%
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\subsubsection{Kleisli to Monoidal}
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For the other direction we are given $(\EndoR, \pure, \bind)$ as in
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For the other direction we are given $(\omapR, \pure, \bind)$ as in
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\ref{eq:monad-kleisli-data} satisfying the laws \kleislilaws. For the data of
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the monoidal formulation we pick:
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%
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\begin{align}
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\begin{split}
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\EndoR & \defeq \EndoRX \\
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\pure & \defeq \pureX \\
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\join & \defeq \bind\ \identity
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\EndoR & \defeq (\omapR, \bind\ (\pure \lll f)) \\
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\pure & \defeq \pure \\
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\join & \defeq \bind\ \identity
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\end{split}
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\end{align}
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%
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Where $\EndoRX \defeq (\bind\ (\pure \lll f), \EndoR)$ and $\pureX \defeq
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\bind\ \identity$. We must now show the laws \monoidallaws, but we must also
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verify that our choice of $\EndoRX$ actually is a functor. I will ommit this
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here. In stead we shall see how these two mappings are indeed inverses.
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We must now not only show the monad laws given for the monoidal
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formulation (\monoidallaws), we must also verify that $\EndoR$ is a
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functor and that $\pure$ and $\join$ are natural transformations. I
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will ommit this here. In stead we shall see how these two mappings are
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indeed inverses. The full construction can be found in the module:
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\begin{center}
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\mbox{\sourcelink{Cat.Category.Monad.Kleisli}}
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\end{center}
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%
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\subsubsection{Equivalence}
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To prove that the two formulations are equivalent we must demonstrate that the
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two mappings sketched above are indeed inverses of each other. If we name the
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first mapping $\toKleisli$ and it's proposed inverse $\toMonoidal$
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then we must show:
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To prove that the two formulations are equivalent we must demonstrate
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that the two mappings sketched above are indeed inverses of each
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other. To recap, these maps are:
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%
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\begin{align*}
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\toKleisli & \tp \var{Kleisli} \to \var{Monoidal} \\
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\toKleisli & \defeq \lambda\ (\omapR, \pure, \bind)
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\to (\EndoR, \pure, \bind\ \identity)
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\end{align*}
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%
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Where $\EndoR \defeq (\omapR, \bind\ (\pure \lll f))$. The proof that
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this is indeed a functor is left implicit as well as the monad laws.
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Likewise the proof that $\pure$ and $\bind\ \identity$ are natural
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transformations are left implicit. The inverse map will be:
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%
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\begin{align*}
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\toMonoidal & \tp \var{Monoidal} \to \var{Kleisli} \\
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\toMonoidal & \defeq \lambda\ (\EndoR, \pureNT, \joinNT)
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\to (\omapR, \pure, \bind)
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\end{align*}
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%
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Where $\bind\ f \defeq \join \lll \fmap\ f$. Again the monad laws are
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left implicit. Now we must show:
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%
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\begin{align}
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\label{eq:monad-forwards}
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@ -1463,30 +1556,37 @@ then we must show:
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\toMonoidal \comp \toKleisli & ≡ \identity
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\end{align}
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%
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For \ref{eq:monad-forwards} let $(\EndoR, \pure, \join)$ be a monad in the
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monoidal form. In my formulation the proof that being-a-monad is a proposition
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can be found. With this result in place we get an equality principle for
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kleisli-monads that say that to equate two such monads it suffices to equate
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their data-part. So it suffices to equate the data-parts of the
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\ref{eq:monad-forwards}. Such a proof is a triple equation the three projections
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of \ref{eq:monad-kleisli-data}. The first two hold definitionally -- essentially
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one just wraps and unwraps the morphism in a functor. For the last equation a
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little more work is required:
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For \ref{eq:monad-forwards} let $(\omapR, \pure, \bind)$ be a monad in
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the Kleisli form. Since being-a-monad is a proposition\footnote{The
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proof can be found in the implementation.} we get an
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equality-principle for kleisli-monads that say that to equate two such
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monads it suffices to equate their data-part. So it suffices to equate
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the data-parts of the \ref{eq:monad-forwards}. Such a proof is a
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triple equating the three projections of \ref{eq:monad-kleisli-data}.
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The first two hold definitionally -- essentially one just wraps and
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unwraps the morphism in a functor. For the last equation a little more
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work is required:
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%
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\begin{align*}
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\join \lll \fmap\ f & ≡
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\fmap\ f \rrr \join \\ & ≡
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\join \lll \fmap\ f & =
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\fmap\ f \rrr \join \\ & =
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\bind\ (f \rrr \pure) \rrr \bind\ \identity
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&& \text{By definition of $\fmap$ and $\join$} \\ & ≡
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\bind\ (f \rrr \pure \fish \identity)
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&& \text{By \ref{eq:monad-kleisli-laws-2}} \\ & ≡
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\bind\ (f \rrr \identity)
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&& \text{By \ref{eq:monad-kleisli-laws-1}} \\ & ≡
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&& \text{By \ref{eq:monad-kleisli-laws-1}} \\ & =
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\bind\ f
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\end{align*}
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%
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For the other direction we can likewise verify that the maps $\EndoR$, $\bind$,
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$\join$, and $\fmap$ are equal. The equality principle for functors gives us
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that this is enough to show that the the functor $\EndoR$ we construct is
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identical. Similarly for the natural transformations we have that the naturality
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condition is a proposition so the paths between the maps are sufficient.
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For the other direction (\ref{eq:monad-backwards}) we are given a
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monad in the monoidal form; $(\EndoR, \pureNT, \joinNT)$. The various
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equality-principles again give us that it is sufficient to equate the
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data-part of the above. That is, we only need to verify that the
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following pieces of data: $\omapR$, $\fmap$, $\pure$ and $\join$ get
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mapped correctly. To see the full details check the implementation in
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the module:
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%
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\begin{center}
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\sourcelink{Cat.Category.Monad}
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\end{center}
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@ -9,8 +9,9 @@
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%% Alternatively:
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%% \newcommand{\defeq}{≔}
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\newcommand{\bN}{\mathbb{N}}
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\newcommand{\bC}{\mathbb{C}}
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\newcommand{\bX}{\mathbb{X}}
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\newcommand{\bC}{ℂ}
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\newcommand{\bD}{𝔻}
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\newcommand{\bX}{𝕏}
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% \newcommand{\to}{\rightarrow}
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%% \newcommand{\mto}{\mapsto}
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\newcommand{\mto}{\rightarrow}
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@ -99,6 +100,10 @@
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\newcommand\Endo[1]{\varindex{Endo}\ #1}
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\newcommand\EndoR{\functor{\mathcal{R}}}
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\newcommand\omapR{\mathcal{R}}
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\newcommand\omapF{\mathcal{F}}
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\newcommand\omapG{\mathcal{G}}
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\newcommand\FunF{\functor{\omapF}}
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\newcommand\FunG{\functor{\omapG}}
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\newcommand\funExt{\varindex{funExt}}
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\newcommand{\suc}[1]{\varindex{suc}\ #1}
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\newcommand{\trans}{\varindex{trans}}
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@ -117,3 +122,5 @@
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}
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\newcommand{\gitlink}{\hrefsymb{\docbasepath}{\texttt{\docbasepath}}}
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\newcommand{\doclink}{\hrefsymb{\sourcebasepath}{\texttt{\sourcebasepath}}}
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\newcommand{\clll}{\mathrel{\bC.\mathord{\lll}}}
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\newcommand{\dlll}{\mathrel{\bD.\mathord{\lll}}}
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@ -140,3 +140,4 @@
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-- (1, 1) -- (1, 0.6);
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}
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}
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\usepackage{ dsfont }
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@ -63,7 +63,8 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
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f ∎
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Kleisli.IsMonad.isDistributive toKleisliIsMonad f g = begin
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bind g >>> bind f ≡⟨⟩
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(join <<< fmap g) >>> (join <<< fmap f) ≡⟨ isDistributive f g ⟩
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(join <<< fmap f) <<< (join <<< fmap g) ≡⟨ isDistributive f g ⟩
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join <<< fmap (join <<< fmap f <<< g) ≡⟨⟩
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bind (g >=> f) ∎
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-- Kleisli.IsMonad.isDistributive toKleisliIsMonad = isDistributive
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Reference in New Issue