Monomorphism

Definition

Let $\mathcal C$ be a category . A morphism $f : X \to Y$ is a monomorphism (or mono) if for every object $Z$ and all morphisms $g_1,g_2 : Z \to X$,

where $\circ$ denotes composition .

Equivalently: $f$ is mono iff it is left-cancellative with respect to composition.

Notes:

• Every isomorphism is a monomorphism.
• The “dual” notion is epimorphism (right-cancellative).

Examples

1. In $\mathbf{Set}$, monomorphisms are exactly injective functions .
2. In $\mathbf{Grp}$, monomorphisms are exactly injective group homomorphisms.
3. In $R\text{-}\mathbf{Mod}$ (and in $\mathbf{Ab}$), monomorphisms are exactly injective module homomorphisms.