Monomorphism - Imported Knowlpedia Content

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 $g_1,g_2 : Z \to X$ ,

f\circ g_1 = f\circ g_2 \quad \Longrightarrow \quad g_1 = g_2,

where $\circ$ denotes composition.

Equivalent characterizations

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

Remarks

Notes:

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

Examples

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