\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*{Kleisli category}

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

\hypertarget{2category_theory}{}\paragraph*{{2-Category theory}}\label{2category_theory}

[[!include 2-category theory - contents]]

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\hypertarget{kleisli_category}{}\section*{{Kleisli category}}\label{kleisli_category}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_terms_of_free_algebras}{In terms of free algebras}\dotfill \pageref*{in_terms_of_free_algebras} \linebreak
\noindent\hyperlink{in_terms_of_kleisli_morphisms}{In terms of Kleisli morphisms}\dotfill \pageref*{in_terms_of_kleisli_morphisms} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{universal_properties}{Universal properties}\dotfill \pageref*{universal_properties} \linebreak
\noindent\hyperlink{in_functional_programming}{In functional programming}\dotfill \pageref*{in_functional_programming} \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}

Given a [[monad]] $T$ on some [[category]] $\mathcal{C}$, then its \emph{Kleisli category} is the [[full subcategory]] of the [[Eilenberg-Moore category]] of $T$, hence the category of [[algebras over a monad|T-algebras]], on those that are \emph{[[free construction|free]]} [[algebra for a monad|T-algebras]] (free $T$-[[modules]]).

Explicitly one may describe the \emph{Kleisli category} of $T$ to have as [[objects]] the objects of $\mathcal{C}$, and a morphism $X \to Y$ in the Kleisli category is a morphism in $\mathcal{C}$ of the form $X \to T(Y)$ in $\mathcal{C}$. The monad structure induces a natural [[composition]] of such ``$T$-shifted'' morphisms.

The Kleisli category is also characterized by the following [[universal property]]:

Since every [[adjunction]] gives rise to a [[monad]] on the [[domain]] of its [[left adjoint]], we might ask if every monad may be construed as arising from an adjunction. This is in fact true, and the [[initial object|initial]] such adjunction in the [[adjoint functor\#RelationToMonads|category of adjunctions]] for the given monad has the Kleisli category as the codomain of its left adjoint.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $\mathbf{T}=(T,\mu,\eta)$ be a [[monad]] in [[Cat]], where $T:C\to C$ is an [[endofunctor]] with multiplication $\mu:T T\to T$ and unit $\eta:Id_C\to T$.

\hypertarget{in_terms_of_free_algebras}{}\subsubsection*{{In terms of free algebras}}\label{in_terms_of_free_algebras}

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{free $\mathbf{T}$-[[algebra|algebra over a monad]]} (or free $\mathbf{T}$-module) is a $\mathbf{T}$-algebra (module) of the form $(T(M),\mu_M)$, where the [[action]] is the component of multiplication transformation $\mu_M : T(T(M))\to T(M)$.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
The \textbf{Kleisli category} $C_{\mathbf{T}}$ of the monad $\mathbf{T}$ the [[subcategory]] of the [[Eilenberg-Moore category]] $C^{\mathbf{T}}$ on the free $\mathbf{T}$-algebras.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
If $U:C^{\mathbf{T}}\to C$ is the [[forgetful functor]] and $F: C\to C^{\mathbf{T}}$ is the [[free functor|free algebra functor]] $F: M\mapsto (T M,\mu_M)$, then the Kleisli category is simply the [[full subcategory]] of $C^{\mathbf{T}}$ containing those objects in the image of $F$.

\end{remark}
\hypertarget{in_terms_of_kleisli_morphisms}{}\subsubsection*{{In terms of Kleisli morphisms}}\label{in_terms_of_kleisli_morphisms}

As another way of looking at this, we can keep the same objects as in $C$ but redefine the morphisms. This was the original Kleisli construction:

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{Kleisli category} $C_{\mathbf{T}}$ has as objects the objects of $C$, and as [[morphisms]] $M\to N$ the elements of the [[hom-set]] $C(M,T(N))$, in other words [[morphisms]] of the form $M \to T(N)$ in $C$, called \textbf{Kleisli morphisms}.

Composition is given by the \textbf{Kleisli composition} rule $g\circ_{Kleisli} f = \mu_P\circ T(g)\circ f$ (as in the [[Grothendieck construction]] (here $M\stackrel{f}\to N\stackrel{g}\to P$).

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
More explicitly, this means that the Kleisli-composite of $f : x \to T y$ with $g : y \to T z$ is the morphism

\begin{displaymath}
x \stackrel{f}{\to} T y 
  \stackrel{T g}{\to}
  T T z \stackrel{\mu z}{\to}
  T z
  \,.
\end{displaymath}
\end{remark}
\begin{proof}
The equivalence between both presentations amounts to the functor $C_{T} \to C^{T}$ being full and faithful. This functor maps any object $X$ to $T(X)$, and any morphism $f \colon X \to T(Y)$ to $T(X) \stackrel{T(f)}{\to} T^2(Y) \stackrel{\mu_Y}{\to} T(Y)$.

Fullness holds because any morphism $g \colon T(X) \to T(Y)$ of algebras has as antecedent the composite $X \stackrel{\eta_X}{\to} T(X) \stackrel{g}{\to} T(Y)$. Indeed, the latter is mapped by the functor into $\mu_Y \circ T(g) \circ T(\eta_X)$, which because $g$ is a morphism of algebras is equal to $g \circ \mu_X \circ T(\eta_X)$, i.e., $g$.

Faithfulness holds as follows: if $\mu_Y \circ T(f) = \mu_Y \circ T(g)$, then precomposing by $\eta_X$ yields $\mu_Y \circ T(f) \circ \eta_X =
\mu_Y \circ \eta_{T(Y)} \circ f = f$ and similarly for $g$, hence $f = g$.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
This Kleisli composition plays an important role in [[computer science]]; for this, see the article at [[monad (in computer science)]].

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

\hypertarget{universal_properties}{}\subsubsection*{{Universal properties}}\label{universal_properties}

In more general 2-categories the [[universal properties]] of [[Kleisli objects]] are dual to the universal properties of [[Eilenberg-Moore category\#Definition|Eilenberg-Moore objects]].

In particular, $C_{\mathbf{T}}$ is [[initial object|initial]] in the [[adjoint functor\#RelationToMonads|category of adjunctions]] for $\mathbf{T}$ (whereas $C^{\mathbf{T}}$ is [[terminal objects|terminal]]). For a proof, see \emph{[[Category Theory in Context]]} Proposition 5.2.12.

\hypertarget{in_functional_programming}{}\subsubsection*{{In functional programming}}\label{in_functional_programming}

In [[type theory|typed]] [[functional programming]], the Kleisli category is used to model [[call-by-value]] functions with side-effects and [[computation]]. See at \emph{[[monad (in computer science)]]} for more on this.

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

\begin{itemize}%
\item [[Kleisli object]]


\item [[Kleisli 2-category]]


\item [[monad (in computer science)]]


\item [[co-Kleisli category]]



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

The original source is

\begin{itemize}%
\item H. Kleisli, \emph{Every standard construction is induced by a pair of adjoint functors} , Proc. Amer. Math. Soc. \textbf{16} (1965) pp.544--546. (\href{http://www.ams.org/journals/proc/1965-016-03/S0002-9939-1965-0177024-4/}{AMS})


\item Jen\"o{} Szigeti, \emph{On limits and colimits in the Kleisli category}, Cahiers de Topologie et G\'e{}om\'e{}trie Diff\'e{}rentielle Cat\'e{}goriques, 24 no. 4 (1983), p. 381-391 (\href{http://www.numdam.org/item?id=CTGDC_1983__24_4_381_0}{NUMDAM})



\end{itemize}
Discussion of cases where the inclusion of the Kleisli category into the [[Eilenberg-Moore category]] is a [[reflective subcategory]] is in

\begin{itemize}%
\item [[Marcelo Fiore]], [[Matias Menni]], \emph{Reflective Kleisli subcategories of the category of Eilenberg-Moore algebras for factorization monads}, Theory and Applications of Categories, Vol. 15, CT2004, No. 2, pp 40-65. (\href{http://www.tac.mta.ca/tac/volumes/15/2/15-02abs.html}{TAC})

\end{itemize}
Discussion in [[internal category]] theory is in

\begin{itemize}%
\item Tomasz Brzeziski, Adrian Vazquez-Marquez, \emph{Internal Kleisli categories}, Journal of Pure and Applied Algebra Volume 215, Issue 9, September 2011, Pages 2135--2147 (\href{http://arxiv.org/abs/0911.4048}{arXiv:0911.4048})

\end{itemize}
Discussion of Kleisli categories in [[type theory]] is in

\begin{itemize}%
\item Alex Simpson, \emph{Recursive types in Kleisli Categories} (\href{http://reference.kfupm.edu.sa/content/r/e/recursive_types_in_kleisli_categories__1738763.pdf}{pdf})

\end{itemize}
[[!redirects Kleisli category]] [[!redirects Kleisli categories]]

[[!redirects Kleisli morphism]] [[!redirects Kleisli function]] [[!redirects Kleisli morphisms]] [[!redirects Kleisli functions]]

[[!redirects Kleisli composition]]



\end{document}