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\begin{document}

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\section*{split coequalizer}

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

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

[[!include infinity-limits - contents]]

\hypertarget{split_coequalizers}{}\section*{{Split coequalizers}}\label{split_coequalizers}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{coequalizers_and_absolute_coequalizers}{Coequalizers and absolute coequalizers}\dotfill \pageref*{coequalizers_and_absolute_coequalizers} \linebreak
\noindent\hyperlink{contractible_pairs}{Contractible pairs}\dotfill \pageref*{contractible_pairs} \linebreak
\noindent\hyperlink{split_epimorphisms}{Split epimorphisms}\dotfill \pageref*{split_epimorphisms} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{BeckCoequalizerForAlgebrasOverAMonad}{Beck coequalizer for algebras over a monad}\dotfill \pageref*{BeckCoequalizerForAlgebrasOverAMonad} \linebreak


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

For purposes of this page, a [[fork]] (some might say a ``cofork'') in a [[category]] $C$ is a diagram of the form

\begin{displaymath}
A \;\underoverset{f}{g}{\rightrightarrows}\; B \overset{e}{\rightarrow} C
\end{displaymath}
such that $e f = e g$. A \textbf{split coequalizer} is a fork together with morphisms $s\colon C\to B$ and $t\colon B\to A$ as below

$\backslash$begin\{center\}$\backslash$begin\{tikzcd\} A $\backslash$arr,shift left,``f'' $\backslash$arr,shift right,``g''' \& B $\backslash$arr,``e'' $\backslash$arl,bend right=40,``t''' \& C $\backslash$arl,bend right=40,``s''' $\backslash$end\{tikzcd\}$\backslash$end\{center\}

such that $e s = 1_C$, $s e = g t$, and $f t = 1_B$. This is equivalent to saying that the morphism $(f,e)\colon g \to e$ has a [[section]] in the [[arrow category]] of $C$.

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

\hypertarget{coequalizers_and_absolute_coequalizers}{}\subsubsection*{{Coequalizers and absolute coequalizers}}\label{coequalizers_and_absolute_coequalizers}

The name ``split coequalizer'' is appropriate, because in any split coequalizer diagram, the morphism $e$ is necessarily a [[coequalizer]] of $f$ and $g$. For given any $h\colon B\to D$ such that $h f = h g$, the composite $h s$ provides a factorization of $h$ through $e$, since $h s e = h g t = h f t = h$, and such a factorization is unique since $e$ is (split) [[epimorphism|epic]]. In fact, a split coequalizer is not just a coequalizer but an [[absolute coequalizer]]: one preserved by all [[functors]].

\hypertarget{contractible_pairs}{}\subsubsection*{{Contractible pairs}}\label{contractible_pairs}

On the other hand, suppose we are given only $f,g\colon A\to B$ and $t\colon B\to A$ such that $f t = 1_B$ and $g t f = g t g$ (which is certainly the case in any split coequalizer, since $g t f = s e f = s e g = g t g$). Such a situation is sometimes called a \textbf{contractible pair}. In this case, any coequalizer of $f$ and $g$ is split, for if $e\colon B\to C$ is a coequalizer of $f$ and $g$, then the equation $g t f = g t g$ implies, by the universal property of $e$, a unique morphism $s\colon C\to B$ such that $s e = g t$, whence $e s e = e g t = e f t = e$ and so $e s = 1_C$ since $e$ is epic.

Similarly, if $e\colon B\to C$ [[split idempotent|splits]] the [[idempotent]] $g t$ with section $s\colon C\to B$, so that $e s = 1$ and $s e = g t$, then we have

\begin{displaymath}
e g = e s e g = e g t g = e g t f = e s e f = e f
\end{displaymath}
and the other identities are obvious; thus $e$ is a split coequalizer of $f$ and $g$.

\hypertarget{split_epimorphisms}{}\subsubsection*{{Split epimorphisms}}\label{split_epimorphisms}

Dually, if $e\colon B\to C$ is a [[split epimorphism]], with a splitting $s\colon C\to B$, say, then $e$ is a split coequalizer of $B \;\underoverset{1}{s e}{\rightrightarrows}\; B$, the morphism $t$ being the identity.

Moreover, $e$ is also the split coequalizer of its [[kernel pair]], if the latter exists. For if $A \;\underoverset{f}{g}{\rightrightarrows}\; B$ is this kernel pair, then the two maps $s e, 1_B \colon B\to B$ satisfy $e \circ s e = e \circ 1_B$, and hence induce a map $t\colon B\to A$ such that $f t = 1_B$ and $g t = s e$.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{BeckCoequalizerForAlgebrasOverAMonad}{}\subsubsection*{{Beck coequalizer for algebras over a monad}}\label{BeckCoequalizerForAlgebrasOverAMonad}

The ``ur-example'' of a split coequalizer is the following. Let $A$ be an [[algebra for a monad|algebra]] for the [[monad]] $T$ on the category $C$, with structure map $a\colon T A \to A$. Then the diagram

\begin{displaymath}
T^2 A \;\underoverset{\mu_A }{T a}{\rightrightarrows}\; T A \overset{a}{\rightarrow} A\, ,
\end{displaymath}
called the [[canonical presentation]] of the algebra $(A,a)$, is a split coequalizer in $C$ (and a [[reflexive coequalizer]] in the [[Eilenberg-Moore category|category of]] $T$-algebras). The splittings are given by $s = \eta_A \colon A \to T A$ and $t = \eta_{T A} \colon T A \to T^2 A$. (Here $\mu$ and $\eta$ are the multiplication and unit of the monad $T$.)

This split coequalizer figures prominently in Beck's [[monadicity theorem]], whence also called the \emph{[[Beck coequalizer]]}.

See also at \emph{\href{Eilenberg-Moore+category#AsColimitCompletionOfKleisliCategory}{Eilenberg-Moore category -- As a colimit completion of the Kleisli category}}.

[[!redirects split coequalizer]] [[!redirects split coequalizers]] [[!redirects split coequaliser]] [[!redirects split coequalisers]] [[!redirects contractible pair]] [[!redirects contractible pairs]]



\end{document}