[[Epimorphism]]
# Split epimorphism

A **split epimorphism** $f : X \to Y$ is a morphism with a **preïnverse** or **section** $s :  Y \to X$, #m/def/cat 
so that $f \circ s = \id_{Y}$.
Note that $f$ is necessarily an [[epimorphism]], and $s$ has the dual property of being a [[Split monomorphism]].
Furthermore $s$ need not be unique.

## Properties

Let $f : X \twoheadrightarrow Y$ be a split epimorphism with section $s : Y \hookrightarrow X$

1. $f$ is the [[Equalizer and coëqualizer|coëqualizer]] of $s f$ and $\id_{X}$, hence it is [[Regular epimorphism|regularly epic]].

See dual [[Split monomorphism#Properties]].

## See also

- [[Extension|Split extension]]
- [[Internal Axiom of Choice]]

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