nLab
monomorphism

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 $f:X\to Y$ such that, equivalently,

• $f$ is an epimorphism in the opposite category $C^{\mathrm{op}}$.

• for any object $A$ the induced morphism $f_{*}:C(A,X)\to C(A,Y)$ is injective.

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

The morphism $f:X\to Y$ is mono precisely if for all $g,h:A\to X$ such that $f_{*}(h):A\stackrel{h}{\to }X\stackrel{f}{\to }Y$ equals $f_{*}(g):A\stackrel{g}{\to }X\stackrel{f}{\to }Y$ we have $g=h$.

Related concepts

• isomorphism classes of monomorphism define subobjects.

• for more details see epimorphism.

• monomorphism in an (∞,1)-category