\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{function monad}

[[!redirects reader monad]]

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

\hypertarget{type_theory}{}\paragraph*{{Type theory}}\label{type_theory}

[[!include type theory - contents]]

\hypertarget{mapping_space}{}\paragraph*{{Mapping space}}\label{mapping_space}

[[!include mapping space - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{algebras}{Algebras}\dotfill \pageref*{algebras} \linebreak
\noindent\hyperlink{relation_to_the_writer_comonad_and_state_monad}{Relation to the writer comonad and state monad}\dotfill \pageref*{relation_to_the_writer_comonad_and_state_monad} \linebreak
\noindent\hyperlink{InTermsOfDependentTypeTheory}{In terms of dependent type theory}\dotfill \pageref*{InTermsOfDependentTypeTheory} \linebreak
\noindent\hyperlink{RelationToRandomVariables}{Relation to random variables in probability theory}\dotfill \pageref*{RelationToRandomVariables} \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}

In a [[cartesian closed category]]/[[type theory]] $\mathcal{C}$, given any [[object]]/[[type]] $W$ there is a [[monad]]

\begin{displaymath}
[W,-] \colon \mathcal{C} \to \mathcal{C}
\end{displaymath}
given by forming the [[internal hom]] out of $W$, hence the ``space of functions'' out of $W$. This is sometimes called the \emph{function monad}. Its [[unit of a monad]] is given by sending values to [[constant functions]] with that value, and the monad operation is given by evaluating on the [[diagonal]].

In the context of [[monads in computer science]] this monad is called the \emph{reader monad} or \emph{environment monad}. It serves to write programs in which all operations may ``read in'' a state of type $W$ (an ``environment'').

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

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

The reader monad does not generally arise from a [[monadic adjunction]]. [[algebra over a monad|Algebras]] may not be easily charcterizable. When $W$ is a 2 element set, algebras correspond to [[idempotent]] [[semigroups]], also known as \emph{rectangular bands}.

\hypertarget{relation_to_the_writer_comonad_and_state_monad}{}\subsubsection*{{Relation to the writer comonad and state monad}}\label{relation_to_the_writer_comonad_and_state_monad}

In a [[cartesian closed category]]/[[type theory]] $\mathcal{C}$, the reader monad $[W,-] \colon \mathcal{C}\to \mathcal{C}$ is [[right adjoint]] to the [[writer comonad]] $W\times (-)$.

Just as the writer comonad is canonically a monad when $W$ is a [[monoid]], so the reader monad is a comonad in that case, and then it is sometimes called the `traced comonad'.

The composite of writer comonad followed by [[reader monad]] is the [[state monad]].

\hypertarget{InTermsOfDependentTypeTheory}{}\subsubsection*{{In terms of dependent type theory}}\label{InTermsOfDependentTypeTheory}

If the type system is even a [[locally Cartesian closed category]]/[[dependent type theory]] then for each type $W$ there is the [[base change]] [[adjoint triple]]

\begin{displaymath}
\mathcal{C}_{/W}
    \stackrel{\overset{\sum_W}{\longrightarrow}}{\stackrel{\overset{W^\ast}{\longleftarrow}}{\underset{\prod_W}{\longrightarrow}}}
  \mathcal{C}
\end{displaymath}
In terms of this then the function monad is equivalently the composite

\begin{displaymath}
\prod_W W^\ast = [W,-] \;\colon\; \mathcal{C} \longrightarrow \mathcal{C}
\end{displaymath}
of [[context extension]] followed by [[dependent product]].

One may also think of this as being the [[polynomial functor]] associated with the span

\begin{displaymath}
\ast \leftarrow W \rightarrow \ast \rightarrow \ast
  \,.
\end{displaymath}
(Notice that the [[comonad]] obtained by composing the other way around, $W^\ast \prod_W$, is the [[modal operator]] usually called \emph{[[necessity]]}.)

\hypertarget{RelationToRandomVariables}{}\subsubsection*{{Relation to random variables in probability theory}}\label{RelationToRandomVariables}

It makes sense to think of $[W,-]$ as producing spaces of [[random variables]] (\hyperlink{Verdier14}{Verdier 14}) depending on [[possible worlds]] $w\in W$ (\hyperlink{TorontoMcCarthy10b}{Toronto-McCarthy 10b, slide 24}).

\begin{quote}%
The intuition behind the Reader monad, for a mathematician, is perhaps stochastic variables. A stochastic variable is a function from a probability space to some other space. So we see a stochastic variable as a monadic value. (\hyperlink{Verdier14}{Verdier 14})

you could interpret this by regarding random variables as reader monad computations. (\hyperlink{TorontoMcCarthy10b}{Toronto-McCarthy 10b, slide 35})


\end{quote}
\hyperlink{TorontoMcCarthy10a}{Toronto-McCarthy 10a, 2.2}, \hyperlink{Toronto14}{Toronto 14, 6.2} call the function monad the \emph{random variable idiom}.

(While, from the perspective of [[modal type theory]], its siblings $W^\ast \prod_W$ and $W^\ast \sum_W$ may be called \emph{[[necessity]]} and \emph{[[possibility]]}, respectively).

In [[Haskell]], the traditional way to model that a program may depend on random states $w \in W$ is to consider not the reader monad but the [[state monad]] $[W,W \times (-)]$. This allows programs not only to depend on the state $w\in W$, but also to change it (for instance to a new number generated by a random number generator). If however one restricts attention to programs that just depend on/read in a random environment $w\in W$ but never change it, then these standard constructs may be reduced from using the state monad to using just the reader monad.

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

\begin{itemize}%
\item [[maybe monad]]


\item [[continuation monad]]


\item [[writer comonad]], [[state monad]]



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

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Monad_%28functional_programming%29#Environment_monad}{Environment monad}}.


\item [[Neil Toronto]], [[Jay McCarthy]], \emph{From Bayesian notation to pure Racket, via measuretheoretic probability} , in \emph{Implementation and Application of Functional Languages}, 2010 (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.298.4274}{web})


\item [[Neil Toronto]], [[Jay McCarthy]], \emph{From Bayesian Notation to Pure Racket}, talk notes 2010 (\href{http://jeapostrophe.github.io/home/static/toronto-2010ifl-slides.pdf}{pdf})


\item [[Neil Toronto]], \emph{Useful Languages for Probabilistic Modeling and Inference}, PhD Thesis, 2014 (\href{http://cs.umd.edu/~ntoronto/papers/toronto-2014diss.pdf}{pdf}, \href{http://cs.umd.edu/~ntoronto/papers/toronto-2014diss-slides.pdf}{slides})


\item [[Olivier Verdier]], \emph{\href{http://www.olivierverdier.com/posts/2014/12/31/reader-writer-monad-comonad/}{The Reader and Writer Monads and Comonads}}, 2014



\end{itemize}
A treatment of \emph{opacity} in [[linguistics]] via the function monad

\begin{itemize}%
\item [[Gianluca Giorgolo]], [[Ash Asudeh]], \emph{Monads as a Solution for Generalized Opacity}, \href{http://users.ox.ac.uk/~cpgl0036/pdf/giorgolo-asudeh-eacl2014.pdf}{pdf}


\item [[Gianluca Giorgolo]], [[Ash Asudeh]], \emph{Perspectives}, Semantics and Pragmatics, vol. 9, \href{http://semprag.org/article/view/sp.9.21}{paper}



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

[[!redirects function monad]] [[!redirects function monads]]

[[!redirects environment monad]] [[!redirects environment monads]]



\end{document}