Monomorphism

Split monomorphism

A split monomorphism π‘š :𝑋 β†’π‘Œ is a morphism with a postinverse or retraction π‘Ÿ :π‘Œ →𝑋, cat so that π‘Ÿ βˆ˜π‘š =id𝑋. Note that π‘š is necessarily a monomorphism, and π‘Ÿ has the dual property of being a split epimorphism. Furthermore π‘Ÿ need not be unique.

Properties

Let π‘š :𝑋 β†ͺπ‘Œ be a split monomorphism with retraction π‘Ÿ :π‘Œ ↠𝑋

  1. π‘š is the equalizer of π‘šπ‘Ÿ and idπ‘Œ, hence it is regularly monic.

See dual properties at Properties

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