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 $𝑟 \circ 𝑚 = i d 𝑋$ . 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 $𝑚 : 𝑋 ↪ 𝑌$ be a split monomorphism with retraction $𝑟 : 𝑌 ↠ 𝑋$

$𝑚$ is the equalizer of $𝑚 𝑟$ and $i d 𝑌$ , hence it is regularly monic.

See dual properties at Properties

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