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\begin{document}

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\section*{split epimorphism}

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

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

[[!include category theory - contents]]

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

\noindent\hyperlink{definitions_and_terminology}{Definitions and terminology}\dotfill \pageref*{definitions_and_terminology} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak


\hypertarget{definitions_and_terminology}{}\subsection*{{Definitions and terminology}}\label{definitions_and_terminology}

A \textbf{split epimorphism} in a [[category]] $C$ is a [[morphism]] $e\colon A \to B$ in $C$ such that there exists a morphism $s\colon B \to A$ such that the [[composite]] $e \circ s$ equals the [[identity morphism]] $1_B$. Then the morphism $s$, which satisfies the [[duality|dual]] condition, is a \textbf{[[split monomorphism]]}.

We say that:

\begin{itemize}%
\item $s$ is a \textbf{[[section]]} of $e$,


\item $e$ is a \textbf{[[retraction]]} of $s$,


\item $B$ is a \textbf{[[retract]]} of $A$,


\item the pair $(e,s)$ is a \textbf{[[split idempotent|splitting]]} of the [[idempotent]] $s \circ e\colon A \to A$.



\end{itemize}
A split epimorphism in $C$ can be equivalently defined as a morphism $e\colon A \to B$ such that for every [[object]] $X\colon C$, the [[function]] $C(X,e)$ is a [[surjection]] in $\mathbf{Set}$; the preimage of $1_B$ under $C(B,e)$ yields a section $s$.

Alternatively, it is also possible to define a split epimorphism as an \textbf{absolute epimorphism}: a morphism such that for every [[functor]] $F$ out of $C$, $F(e)$ is an [[epimorphism]]. From the definition as a morphism having a section, it is obvious that any split epimorphism is absolute; conversely, that the image of $e$ under the representable functor $C(B,1)$ is an epimorphism reduces to the characterization above.

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

\begin{itemize}%
\item Any split epimorphism is automatically a [[regular epimorphism]] (it is the [[coequalizer]] of $s\circ e$ and $1_A$), and therefore also a [[strong epimorphism]], an [[extremal epimorphism]], and (of course) an [[epimorphism]].


\item The [[axiom of choice]] [[internal logic|internal]] to a category $C$ can be phrased as ``all epimorphisms are split.'' In [[Set]] this is equivalent to the usual axiom of choice; in many other categories it may be true without assuming the axiom of choice (in $Set$), or it may be false regardless of the axiom of choice.



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

The notion of split epimorphism arises often as a condition on fibrations in [[categories of chain complexes]]. See there for details.

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

\begin{itemize}%
\item In [[Vect]], every epimorphism is split. For $\phi\colon V \to W$ a surjective linear map, we can find an isomorphism $V \simeq ker(\phi) \oplus V'$. Then $\phi|_{V'}$ is an isomorphism, and its inverse $W \to V' \hookrightarrow ker(\phi) \oplus V'$ is a section of $\phi$.

\end{itemize}
[[!redirects split epimorphism]] [[!redirects split epimorphisms]] [[!redirects split epi]] [[!redirects split epis]] [[!redirects split epic]]

[[!redirects absolute epimorphism]] [[!redirects absolute epimorphisms]] [[!redirects absolute epi]] [[!redirects absolute epis]] [[!redirects absolute epic]]



\end{document}