[[Monomorphism]]
# Split monomorphism

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

## Properties

Let $m : X \hookrightarrow Y$ be a split monomorphism with retraction $r :  Y \twoheadrightarrow X$

1. $m$ is the [[Equalizer and coëqualizer|equalizer]] of $mr$ and $\id_{Y}$, hence it is [[Regular monomorphism|regularly monic]].

See dual properties at [[Split epimorphism#Properties]]

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