nLab
monomorphism

Contents

Idea

The notion of monomorphism is the generalization of the notion of injective map of sets from the category Set to arbitrary categories.

Definition

A monomorphism in a category C is a morphism f:XY such that, equivalently,

The last condtition here states the usual arrow-theoretic way to say monomorphism:

The morphism f:XY is mono precisely if for all g,h:AX such that f *(h):AhXfY equals f *(g):AgXfY we have g=h.