\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*{reflexive coequalizer}

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

\hypertarget{limits_and_colimits}{}\paragraph*{{Limits and colimits}}\label{limits_and_colimits}

[[!include infinity-limits - contents]]

\hypertarget{reflexive_coequalisers}{}\section*{{Reflexive coequalisers}}\label{reflexive_coequalisers}

\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{reflexive pair} is a [[parallel pair]] $f,g\colon A\rightrightarrows B$ having a common [[section]], i.e. a map $s\colon B\to A$ such that $f \circ s = g \circ s = 1_B$. A \textbf{reflexive coequalizer} is a [[coequalizer]] of a reflexive pair. A category \textbf{has reflexive coequalizers} if it has coequalizers of all reflexive pairs.

Dually, a reflexive coequalizer in the [[opposite category]] $C^{op}$ is called a \textbf{coreflexive equalizer} in $C$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Reflexive coequalizers should not be confused with [[split coequalizers]], a distinct concept.

\end{remark}
\begin{example}
\label{}\hypertarget{}{}
Any [[congruence]] is a reflexive pair, so in particular any [[quotient object|quotient]] of a congruence is a reflexive coequalizer.

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

\begin{theorem}
\label{LintonTheorem}\hypertarget{LintonTheorem}{}
If $T$ is a [[monad]] on a [[cocomplete category]] $C$, then the category $C^T$ of [[Eilenberg-Moore category|Eilenberg Moore algebras]] is cocomplete if and only if it has reflexive coequalizers. This is the case particularly if $T$ preserves reflexive coequalizers.

\end{theorem}
This is due to (\hyperlink{Linton}{Linton}).

\begin{proof}
Suppose $C^T$ has reflexive coequalizers. Then $C^T$ certainly has coproducts, because if $A_i$ is a collection of $T$-algebras, then we can form the coequalizer in $C^T$ of the reflexive pair

\begin{displaymath}
\sum_i F U F U A_i \stackrel{\overset{\sum_i \varepsilon F U A_i}{\to}}{\underset{\sum_i F U \varepsilon A_i}{\to}} \sum_i F U A_i
\end{displaymath}
using the fact that the displayed coproducts exist because, for example,

\begin{displaymath}
\sum_i F U A_i \cong F(\sum_i U A_i)
\end{displaymath}
since the left adjoint $F$ preserves coproducts, assumed to exist in $C$. That this reflexive coequalizer is the coproduct $\sum_i A_i$ in $C^T$ is routine.

Finally, a category with coproducts and reflexive coequalizers is cocomplete. It suffices that general coequalizers exist, but it is easily seen that if

\begin{displaymath}
f, g \colon A \stackrel{\to}{\to} B
\end{displaymath}
is a parallel pair, then the coequalizer of the reflexive pair

\begin{displaymath}
A + B \stackrel{\overset{(f, 1_B)}{\to}}{\underset{(g, 1_B)}{\to}} B
\end{displaymath}
(note both maps are retracts of the inclusion $B \to A + B$) also exists, and gives the coequalizer of the first pair. This completes the proof.

\end{proof}
\begin{prop}
\label{PreservingReflectiveCoequalizersInTwoVariables}\hypertarget{PreservingReflectiveCoequalizersInTwoVariables}{}
If $F\colon C\times D\to E$ is a [[functor]] of two variables which preserves reflexive coequalizers in each variable separately (that is, $F(c,-)$ and $F(-,d)$ preserve reflexive coequalizers for all $c\in C$ and $d\in D$), then $F$ preserves reflexive coequalizers in both variables together.

\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
This is emphatically \emph{not} the case for arbitrary coequalizers.

\end{remark}
\begin{proof}
\textbf{of proposition \ref{PreservingReflectiveCoequalizersInTwoVariables}} Suppose given two reflexive coequalizers

\begin{displaymath}
c_0 \stackrel{\to}{\to} c_1 \to c_2
\end{displaymath}
\begin{displaymath}
\,
\end{displaymath}
\begin{displaymath}
d_0 \stackrel{\to}{\to} d_1 \to d_2
\end{displaymath}
and let $c_{i j}$ denote $F(c_i, d_j)$ so that we have a diagram

\begin{displaymath}
\itexarray{
c_{0 0} & \stackrel{\to}{\to} & c_{0 1} & \to & c_{0 2} \\
\downarrow \downarrow & & \downarrow \downarrow & & \downarrow \downarrow \\
c_{1 0} & \stackrel{\to}{\to} & c_{1 1} & \to & c_{1 2} \\ 
\downarrow & & \downarrow & & \downarrow \\
c_{2 0} & \stackrel{\to}{\to} & c_{2 1} & \to & c_{2 2} 
}
\end{displaymath}
in which all rows and columns are reflexive coequalizers (using preservation of reflexive coequalizers in separate variables), and all squares are \emph{serially} commutative. According to \hyperlink{BarrWells}{Toposes, Triples, Theories}, lemma 4.2 page 248, the diagonal is also a (reflexive) coequalizer, as claimed. (See also the lemma on page 1 of Johnstone's \hyperlink{Johnstone}{Topos Theory}.)

\end{proof}
Proposition \ref{PreservingReflectiveCoequalizersInTwoVariables} is particularly interesting when $F$ is the [[tensor product]] of a cocomplete [[closed monoidal category]] $C$. In this case it implies that the [[free monoid monad]] on such a category preserves reflexive coequalizers, and thus (by \hyperlink{LintonTheorem}{Linton's theorem}) the category of [[monoid objects]] in $C$ is cocomplete.

\begin{prop}
\label{}\hypertarget{}{}
Reflexive coequalizers in [[Set]] commute with finite [[products]]:

the $n$-fold product functors $Set^n \stackrel{\prod}{\to} Set$ preserve reflexive coequalizers.

\end{prop}
\begin{proof}
This follows from prop. \ref{PreservingReflectiveCoequalizersInTwoVariables} as well as from the fact that the [[diagram]] category $\{ 0 \stackrel{\overset{d_0}{\to}}{\stackrel{\overset{s_0}{\leftarrow}}{\underset{d_1}{\to}}} 1\}$ with $d_0 \circ s_0 = d_1 \circ s_0 = id$ is a [[sifted category]].

\end{proof}
Of course, the diagonal functor $\Delta: Set \to Set^n$, being [[left adjoint]] to the product functor, preserves reflexive coequalizers; therefore the composite

\begin{displaymath}
Set \stackrel{\prod \Delta}{\to} Set: x \mapsto \hom(n, x)
\end{displaymath}
also preserves reflexive coequalizers.

This has a further consequence which is technically very convenient:

\begin{theorem}
\label{}\hypertarget{}{}
If $T$ is a [[finitary monad]] on $Set$, then $T$ preserves reflexive coequalizers.

\end{theorem}
\begin{proof}
We have a [[coend]] formula for $T$:

\begin{displaymath}
T(-) \cong \int^{n \in Fin} T(n) \times \hom(n, -)
\end{displaymath}
and since this is a colimit of functors $\hom(n, -)$ which preserve reflexive coequalizers, $T$ must also preserve reflexive coequalizers.

\end{proof}
Since finitary monads $T$ preserve reflexive coequalizers, it follows that the monadic functor $U \colon Set^T \to Set$ reflects reflexive coequalizers, and so since $Set$ has reflexive coequalizers, $Set^T$ must as well. Therefore, by proposition \ref{LintonTheorem}, $Set^T$ is cocomplete. This is actually true for infinitary monads $T$ on $Set$ as well, at least if we assume the axiom of choice (see \href{http://ncatlab.org/nlab/show/colimits+in+categories+of+algebras#reflexive_coequalizers_and_cocompleteness_16}{here} for a proof), but the argument just given is a choice-free proof for the case of finitary monads.

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\begin{itemize}%
\item Reflexive coequalizers figure in the [[crude monadicity theorem]].

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

\begin{itemize}%
\item [[Fred Linton]], \emph{Coequalizers in categories of algebras}, Seminar on Triples and Categorical Homology Theory, Lecture Notes in Mathematics Vol. 80 (1969), 75-90.

\end{itemize}
\begin{itemize}%
\item [[Michael Barr]] and [[Charles Wells]], \emph{Toposes, Theories, and Triples}, Reprints in Theory and Applications of Categories (2005), 1-289. (\href{http://www.case.edu/artsci/math/wells/pub/pdf/ttt.pdf}{online pdf})

\end{itemize}
\begin{itemize}%
\item [[Peter Johnstone]], Topos Theory, London Mathematical Society Monographs no. 10, Academic Press, 1977.

\end{itemize}
[[!redirects reflexive coequalizer]] [[!redirects reflexive coequalizers]] [[!redirects reflexive coequaliser]] [[!redirects reflexive coequalisers]] [[!redirects coreflexive equalizer]] [[!redirects coreflexive equalizers]] [[!redirects coreflexive equaliser]] [[!redirects coreflexive equalisers]] [[!redirects reflexive equalizer]] [[!redirects reflexive equalizers]] [[!redirects reflexive equaliser]] [[!redirects reflexive equalisers]]



\end{document}