Add section on functors and natural transformations
Also do not use ugly overbar
This commit is contained in:
parent
2fce963072
commit
01159930de
|
|
@ -1261,6 +1261,55 @@ That in any category:
|
|||
\prod_{A\ B \tp \Object} \isProp\ (\var{Product}\ \bC\ A\ B)
|
||||
\end{align}
|
||||
%
|
||||
\section{Functors and natural transformations}
|
||||
For the sake of completeness I will mention the definition of functors
|
||||
and natural transformations. Please refer to the implementation for
|
||||
the full details.
|
||||
%
|
||||
\subsection{Functors}
|
||||
Given two categories $\bC$ and $\bD$ a functor consists of the
|
||||
following data:
|
||||
%
|
||||
\begin{align*}
|
||||
\omapF & \tp ℂ.\Object → 𝔻.\Object \\
|
||||
\fmap & \tp ℂ.\Arrow\ A\ B → 𝔻.\Arrow\ (\omapF\ A)\ (\omapF\ B)
|
||||
\end{align*}
|
||||
%
|
||||
And the following laws:
|
||||
\begin{align*}
|
||||
\fmap\ ℂ.\identity & ≡ 𝔻.identity \\
|
||||
\fmap\ (g \clll f) & ≡ \fmap\ g \dlll \fmap\ f
|
||||
\end{align*}
|
||||
%
|
||||
The implementation can be found here:
|
||||
%
|
||||
\begin{center}
|
||||
\sourcelink{Cat.Category.Functor}
|
||||
\end{center}
|
||||
\subsection{Natural Transformation}
|
||||
Given two functors between categories $\bC$ and $\bD$. Name them
|
||||
$\FunF$ and $\FunG$. A natural transformation is a family of arrows:
|
||||
%
|
||||
\begin{align*}
|
||||
\prod_{C \tp ℂ.\Object} \bD.\Arrow\ (\omapF\ C)\ (\omapG\ C)
|
||||
\end{align*}
|
||||
%
|
||||
This family of arrows can be seen as the data. If $\theta$ is a
|
||||
natural transformation $\theta\ C$ will be called the component (of
|
||||
$\theta$) at $C$. The laws of this family of morphism is the
|
||||
naturality condition:
|
||||
%
|
||||
\begin{align*}
|
||||
\prod_{f \tp ℂ.\Arrow\ A\ B}
|
||||
(θ\ B) \dlll (\FunF.\fmap\ f) ≡ (\FunG.\fmap\ f) \dlll (θ\ A)
|
||||
\end{align*}
|
||||
%
|
||||
The implementation can be found here:
|
||||
%
|
||||
\begin{center}
|
||||
\sourcelink{Cat.Category.NaturalTransformation}
|
||||
\end{center}
|
||||
|
||||
\section{Monads}
|
||||
\label{sec:monads}
|
||||
In this section I present two formulations of monads. The two representations
|
||||
|
|
@ -1269,8 +1318,14 @@ simply monoidal monads and Kleisli monads for short. We then show that the two
|
|||
formulations are equivalent, which due to univalence gives us a path between the
|
||||
two types.
|
||||
|
||||
Let a category $\bC$ be given. In the remainder of this sections all objects and
|
||||
arrows will implicitly refer to objects and arrows in this category.
|
||||
Let a category $\bC$ be given. In the remainder of this sections all
|
||||
objects and arrows will implicitly refer to objects and arrows in this
|
||||
category. I will also use the notation $\EndoR$ to refer to an
|
||||
endofunctor on this category. Its map on objects will be denoted
|
||||
$\omapR$ and its map on arrows will be denoted $\fmap$. Likewise I
|
||||
will use the notation $\pureNT$ to refer to a natural transformation
|
||||
and its component at a given (implicit) object will be denoted
|
||||
$\pure$.
|
||||
%
|
||||
\subsection{Monoidal formulation}
|
||||
The monoidal formulation of monads consists of the following data:
|
||||
|
|
@ -1278,7 +1333,7 @@ The monoidal formulation of monads consists of the following data:
|
|||
\begin{align}
|
||||
\label{eq:monad-monoidal-data}
|
||||
\begin{split}
|
||||
\EndoR & \tp \Endo ℂ \\
|
||||
\EndoR & \tp \Functor\ ℂ\ \bC \\
|
||||
\pureNT & \tp \NT{\EndoR^0}{\EndoR} \\
|
||||
\joinNT & \tp \NT{\EndoR^2}{\EndoR}
|
||||
\end{split}
|
||||
|
|
@ -1321,8 +1376,15 @@ The Kleisli-formulation consists of the following data:
|
|||
\end{split}
|
||||
\end{align}
|
||||
%
|
||||
The objects $X$ and $Y$ are implicitly universally quantified.
|
||||
|
||||
The objects $X$ and $Y$ are implicitly universally quantified. With this data we can construct the \nomenindex{Kleisli arrow}:
|
||||
%
|
||||
\begin{align*}
|
||||
\fish & \tp \Arrow\ A\ (\omapR\ B)
|
||||
\to \Arrow\ B\ (\omapR\ C)
|
||||
\to \Arrow\ A\ (\omapR\ C) \\
|
||||
f \fish g & \defeq f \rrr (\bind\ g)
|
||||
\end{align*}
|
||||
%
|
||||
It is interesting to note here that this formulation does not talk about natural
|
||||
transformations or other such constructs from category theory. All we have here
|
||||
is a regular maps on objects and a pair of arrows.
|
||||
|
|
@ -1339,11 +1401,10 @@ This data must satisfy:
|
|||
\end{align}
|
||||
\newcommand\kleislilaws{\ref{eq:monad-kleisli-laws-0}, \ref{eq:monad-kleisli-laws-1} and \ref{eq:monad-kleisli-laws-2}}%
|
||||
%
|
||||
Here likewise the arrows $f \tp \Arrow\ X\ (\EndoR\ Y)$ and $g \tp
|
||||
\Arrow\ Y\ (\EndoR\ Z)$ are universally quantified (as well as the objects they
|
||||
range over). $\fish$ is the Kleisli-arrow which is defined as $f \fish g \defeq
|
||||
f \rrr (\bind\ g)$ . (\TODO{Better way to typeset $\fish$?})
|
||||
|
||||
Here likewise the arrows $f \tp \Arrow\ X\ (\omapR\ Y)$ and $g \tp
|
||||
\Arrow\ Y\ (\omapR\ Z)$ are universally quantified as well as the
|
||||
objects they range over.
|
||||
%
|
||||
\subsection{Equivalence of formulations}
|
||||
%
|
||||
The notation I have chosen here in the report
|
||||
|
|
@ -1376,27 +1437,27 @@ show that $(\omapR, \pure, \bind)$ is indeed a monad in the Kleisli
|
|||
form. In the second part we will show the other direction.
|
||||
|
||||
\subsubsection{Monoidal to Kleisli}
|
||||
Let $(\EndoR, \pure, \join)$ be given as in \ref{eq:monad-monoidal-data}
|
||||
Let $(\EndoR, \pureNT, \joinNT)$ be given as in \ref{eq:monad-monoidal-data}
|
||||
satisfying the laws \monoidallaws. For the data of the Kleisli
|
||||
formulation we pick:
|
||||
%
|
||||
\begin{align}
|
||||
\begin{split}
|
||||
\EndoR & \defeq \EndoRX \\
|
||||
\pure & \defeq \pureX \\
|
||||
\bind\ f & \tp \joinX \lll \fmapX\ f
|
||||
\omapR & \defeq \omapR \\
|
||||
\pure & \defeq \pure \\
|
||||
\bind\ f & \defeq \join \lll \fmap\ f
|
||||
\end{split}
|
||||
\end{align}
|
||||
%
|
||||
$\EndoRX$ is the object map of the endo-functor $\EndoR$, $\pureX$ and
|
||||
$\joinX$ are the arrows from the natural transformations $\pure$ and
|
||||
$\join$ respectively. The term $\fmapX$ is the arrow map of the
|
||||
endo-functor $\EndoR$. It now just remains to verify the laws
|
||||
Again $\omapR$ is the object map of the endo-functor $\EndoR$, $\pure$
|
||||
and $\join$ are the arrows from the natural transformations $\pureNT$
|
||||
and $\joinNT$ respectively and $\fmap$ is the map on arrows of the
|
||||
endofunctor $\EndoR$. It now just remains to verify the laws
|
||||
\kleislilaws. For \ref{eq:monad-kleisli-laws-0}:
|
||||
%
|
||||
\begin{align*}
|
||||
\bind\ \pure & ≡
|
||||
\join \lll (\fmap\ \pure) && \text{By definition} \\
|
||||
\bind\ \pure & =
|
||||
\join \lll (\fmap\ \pure) \\
|
||||
& ≡ \identity && \text{By \ref{eq:monad-monoidal-laws-2}}
|
||||
\end{align*}
|
||||
%
|
||||
|
|
@ -1404,57 +1465,89 @@ For \ref{eq:monad-kleisli-laws-1}:
|
|||
%
|
||||
\begin{align*}
|
||||
\pure \fish f
|
||||
& \equiv %%%
|
||||
\pure \ggg \bind\ f && \text{By definition} \\ & ≡
|
||||
\bind\ f \lll \pure && \text{By definition} \\ & ≡
|
||||
\joinX \lll \fmapX\ f \lll \pureX && \text{By definition} \\ & ≡
|
||||
\joinX \lll \pureX \lll f && \text{$\pure$ is a natural transformation} \\ & ≡
|
||||
& = %%%
|
||||
\pure \ggg \bind\ f \\ & =
|
||||
\bind\ f \lll \pure \\ & =
|
||||
\join \lll \fmap\ f \lll \pure \\ & ≡
|
||||
\join \lll \pure \lll f && \text{$\pure$ is a natural transformation} \\ & ≡
|
||||
\identity \lll f && \text{By \ref{eq:monad-monoidal-laws-1}} \\ & ≡
|
||||
f && \text{Left identity}
|
||||
\end{align*}
|
||||
%
|
||||
For \ref{eq:monad-kleisli-laws-2}:
|
||||
\begin{align*}
|
||||
\bind\ g \rrr \bind\ f & ≡
|
||||
\bind\ g \rrr \bind\ f & =
|
||||
\bind\ f \lll \bind\ g
|
||||
\\ & ≡
|
||||
\\ & =
|
||||
%% %%%%
|
||||
\joinX \lll \fmapX\ g \lll \joinX \lll \fmapX\ f
|
||||
&& \text{\dots} \\ & ≡
|
||||
\joinX \lll \joinX \lll \fmapX^2\ g \lll \fmapX\ f
|
||||
\join \lll \fmap\ g \lll \join \lll \fmap\ f
|
||||
\\ & ≡
|
||||
\join \lll \join \lll (\fmap \comp \fmap)\ f \lll \fmap\ g
|
||||
&& \text{$\join$ is a natural transformation} \\ & ≡
|
||||
\joinX\ \lll \fmapX\ \joinX \lll \fmapX^2\ g \lll \fmapX\ f
|
||||
\join \lll \fmap\ \join \lll (\fmap \comp \fmap)\ f \lll \fmap\ g
|
||||
&& \text{By \ref{eq:monad-monoidal-laws-0}} \\ & ≡
|
||||
\joinX\ \lll \fmapX\ \joinX\ \lll \fmapX\ (\fmapX\ g) \lll \fmapX\ f
|
||||
\join \lll \fmap\ \join \lll \fmap\ (\fmap\ f) \lll \fmap\ g
|
||||
&& \text{} \\ & ≡
|
||||
\joinX \lll \fmapX\ (\joinX \lll \fmapX\ g \lll f)
|
||||
&& \text{Distributive law for functors} \\ & \equiv
|
||||
\join \lll \fmap\ (\join \lll \fmap\ f \lll g)
|
||||
&& \text{Distributive law for functors} \\ & =
|
||||
\join \lll \fmap\ (\join \lll \fmap\ f \lll g) \\ & =
|
||||
%%%%
|
||||
\bind\ (\bind\ f \lll g) \\ & =
|
||||
\bind\ (g \rrr \bind\ f) \\ & =
|
||||
\bind\ (g \fish f)
|
||||
\end{align*}
|
||||
%
|
||||
The construction can be found in the module:
|
||||
\begin{center}
|
||||
\sourcelink{Cat.Category.Monad.Monoidal}
|
||||
\end{center}
|
||||
%
|
||||
\subsubsection{Kleisli to Monoidal}
|
||||
For the other direction we are given $(\EndoR, \pure, \bind)$ as in
|
||||
For the other direction we are given $(\omapR, \pure, \bind)$ as in
|
||||
\ref{eq:monad-kleisli-data} satisfying the laws \kleislilaws. For the data of
|
||||
the monoidal formulation we pick:
|
||||
%
|
||||
\begin{align}
|
||||
\begin{split}
|
||||
\EndoR & \defeq \EndoRX \\
|
||||
\pure & \defeq \pureX \\
|
||||
\EndoR & \defeq (\omapR, \bind\ (\pure \lll f)) \\
|
||||
\pure & \defeq \pure \\
|
||||
\join & \defeq \bind\ \identity
|
||||
\end{split}
|
||||
\end{align}
|
||||
%
|
||||
Where $\EndoRX \defeq (\bind\ (\pure \lll f), \EndoR)$ and $\pureX \defeq
|
||||
\bind\ \identity$. We must now show the laws \monoidallaws, but we must also
|
||||
verify that our choice of $\EndoRX$ actually is a functor. I will ommit this
|
||||
here. In stead we shall see how these two mappings are indeed inverses.
|
||||
|
||||
We must now not only show the monad laws given for the monoidal
|
||||
formulation (\monoidallaws), we must also verify that $\EndoR$ is a
|
||||
functor and that $\pure$ and $\join$ are natural transformations. I
|
||||
will ommit this here. In stead we shall see how these two mappings are
|
||||
indeed inverses. The full construction can be found in the module:
|
||||
\begin{center}
|
||||
\mbox{\sourcelink{Cat.Category.Monad.Kleisli}}
|
||||
\end{center}
|
||||
%
|
||||
\subsubsection{Equivalence}
|
||||
To prove that the two formulations are equivalent we must demonstrate that the
|
||||
two mappings sketched above are indeed inverses of each other. If we name the
|
||||
first mapping $\toKleisli$ and it's proposed inverse $\toMonoidal$
|
||||
then we must show:
|
||||
To prove that the two formulations are equivalent we must demonstrate
|
||||
that the two mappings sketched above are indeed inverses of each
|
||||
other. To recap, these maps are:
|
||||
%
|
||||
\begin{align*}
|
||||
\toKleisli & \tp \var{Kleisli} \to \var{Monoidal} \\
|
||||
\toKleisli & \defeq \lambda\ (\omapR, \pure, \bind)
|
||||
\to (\EndoR, \pure, \bind\ \identity)
|
||||
\end{align*}
|
||||
%
|
||||
Where $\EndoR \defeq (\omapR, \bind\ (\pure \lll f))$. The proof that
|
||||
this is indeed a functor is left implicit as well as the monad laws.
|
||||
Likewise the proof that $\pure$ and $\bind\ \identity$ are natural
|
||||
transformations are left implicit. The inverse map will be:
|
||||
%
|
||||
\begin{align*}
|
||||
\toMonoidal & \tp \var{Monoidal} \to \var{Kleisli} \\
|
||||
\toMonoidal & \defeq \lambda\ (\EndoR, \pureNT, \joinNT)
|
||||
\to (\omapR, \pure, \bind)
|
||||
\end{align*}
|
||||
%
|
||||
Where $\bind\ f \defeq \join \lll \fmap\ f$. Again the monad laws are
|
||||
left implicit. Now we must show:
|
||||
%
|
||||
\begin{align}
|
||||
\label{eq:monad-forwards}
|
||||
|
|
@ -1463,30 +1556,37 @@ then we must show:
|
|||
\toMonoidal \comp \toKleisli & ≡ \identity
|
||||
\end{align}
|
||||
%
|
||||
For \ref{eq:monad-forwards} let $(\EndoR, \pure, \join)$ be a monad in the
|
||||
monoidal form. In my formulation the proof that being-a-monad is a proposition
|
||||
can be found. With this result in place we get an equality principle for
|
||||
kleisli-monads that say that to equate two such monads it suffices to equate
|
||||
their data-part. So it suffices to equate the data-parts of the
|
||||
\ref{eq:monad-forwards}. Such a proof is a triple equation the three projections
|
||||
of \ref{eq:monad-kleisli-data}. The first two hold definitionally -- essentially
|
||||
one just wraps and unwraps the morphism in a functor. For the last equation a
|
||||
little more work is required:
|
||||
For \ref{eq:monad-forwards} let $(\omapR, \pure, \bind)$ be a monad in
|
||||
the Kleisli form. Since being-a-monad is a proposition\footnote{The
|
||||
proof can be found in the implementation.} we get an
|
||||
equality-principle for kleisli-monads that say that to equate two such
|
||||
monads it suffices to equate their data-part. So it suffices to equate
|
||||
the data-parts of the \ref{eq:monad-forwards}. Such a proof is a
|
||||
triple equating the three projections of \ref{eq:monad-kleisli-data}.
|
||||
The first two hold definitionally -- essentially one just wraps and
|
||||
unwraps the morphism in a functor. For the last equation a little more
|
||||
work is required:
|
||||
%
|
||||
\begin{align*}
|
||||
\join \lll \fmap\ f & ≡
|
||||
\fmap\ f \rrr \join \\ & ≡
|
||||
\join \lll \fmap\ f & =
|
||||
\fmap\ f \rrr \join \\ & =
|
||||
\bind\ (f \rrr \pure) \rrr \bind\ \identity
|
||||
&& \text{By definition of $\fmap$ and $\join$} \\ & ≡
|
||||
\bind\ (f \rrr \pure \fish \identity)
|
||||
&& \text{By \ref{eq:monad-kleisli-laws-2}} \\ & ≡
|
||||
\bind\ (f \rrr \identity)
|
||||
&& \text{By \ref{eq:monad-kleisli-laws-1}} \\ & ≡
|
||||
&& \text{By \ref{eq:monad-kleisli-laws-1}} \\ & =
|
||||
\bind\ f
|
||||
\end{align*}
|
||||
%
|
||||
For the other direction we can likewise verify that the maps $\EndoR$, $\bind$,
|
||||
$\join$, and $\fmap$ are equal. The equality principle for functors gives us
|
||||
that this is enough to show that the the functor $\EndoR$ we construct is
|
||||
identical. Similarly for the natural transformations we have that the naturality
|
||||
condition is a proposition so the paths between the maps are sufficient.
|
||||
For the other direction (\ref{eq:monad-backwards}) we are given a
|
||||
monad in the monoidal form; $(\EndoR, \pureNT, \joinNT)$. The various
|
||||
equality-principles again give us that it is sufficient to equate the
|
||||
data-part of the above. That is, we only need to verify that the
|
||||
following pieces of data: $\omapR$, $\fmap$, $\pure$ and $\join$ get
|
||||
mapped correctly. To see the full details check the implementation in
|
||||
the module:
|
||||
%
|
||||
\begin{center}
|
||||
\sourcelink{Cat.Category.Monad}
|
||||
\end{center}
|
||||
|
|
|
|||
|
|
@ -9,8 +9,9 @@
|
|||
%% Alternatively:
|
||||
%% \newcommand{\defeq}{≔}
|
||||
\newcommand{\bN}{\mathbb{N}}
|
||||
\newcommand{\bC}{\mathbb{C}}
|
||||
\newcommand{\bX}{\mathbb{X}}
|
||||
\newcommand{\bC}{ℂ}
|
||||
\newcommand{\bD}{𝔻}
|
||||
\newcommand{\bX}{𝕏}
|
||||
% \newcommand{\to}{\rightarrow}
|
||||
%% \newcommand{\mto}{\mapsto}
|
||||
\newcommand{\mto}{\rightarrow}
|
||||
|
|
@ -99,6 +100,10 @@
|
|||
\newcommand\Endo[1]{\varindex{Endo}\ #1}
|
||||
\newcommand\EndoR{\functor{\mathcal{R}}}
|
||||
\newcommand\omapR{\mathcal{R}}
|
||||
\newcommand\omapF{\mathcal{F}}
|
||||
\newcommand\omapG{\mathcal{G}}
|
||||
\newcommand\FunF{\functor{\omapF}}
|
||||
\newcommand\FunG{\functor{\omapG}}
|
||||
\newcommand\funExt{\varindex{funExt}}
|
||||
\newcommand{\suc}[1]{\varindex{suc}\ #1}
|
||||
\newcommand{\trans}{\varindex{trans}}
|
||||
|
|
@ -117,3 +122,5 @@
|
|||
}
|
||||
\newcommand{\gitlink}{\hrefsymb{\docbasepath}{\texttt{\docbasepath}}}
|
||||
\newcommand{\doclink}{\hrefsymb{\sourcebasepath}{\texttt{\sourcebasepath}}}
|
||||
\newcommand{\clll}{\mathrel{\bC.\mathord{\lll}}}
|
||||
\newcommand{\dlll}{\mathrel{\bD.\mathord{\lll}}}
|
||||
|
|
|
|||
|
|
@ -140,3 +140,4 @@
|
|||
-- (1, 1) -- (1, 0.6);
|
||||
}
|
||||
}
|
||||
\usepackage{ dsfont }
|
||||
|
|
|
|||
|
|
@ -63,7 +63,8 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
|||
f ∎
|
||||
Kleisli.IsMonad.isDistributive toKleisliIsMonad f g = begin
|
||||
bind g >>> bind f ≡⟨⟩
|
||||
(join <<< fmap g) >>> (join <<< fmap f) ≡⟨ isDistributive f g ⟩
|
||||
(join <<< fmap f) <<< (join <<< fmap g) ≡⟨ isDistributive f g ⟩
|
||||
join <<< fmap (join <<< fmap f <<< g) ≡⟨⟩
|
||||
bind (g >=> f) ∎
|
||||
-- Kleisli.IsMonad.isDistributive toKleisliIsMonad = isDistributive
|
||||
|
||||
|
|
|
|||
Loading…
Reference in a new issue