nLab effective monomorphism

Idea

• effective epimorphism$\Rightarrow$ regular epimorphism $\Leftrightarrow$ covering

• effective monomorphism$\Rightarrow$ regular monomorphism $\Leftrightarrow$ embedding .

Definition

A morphism $f : X \to Y$ in a category $C$ is an effective monomorphism if

1. it has a cokernel pair, i.e. if the pushout $Y \coprod_X Y$ exists;

2. it is the equalizer of the canonical pair of morphisms $Y \stackrel{\to}{\to}Y \coprod_X Y$.

Such a morphism, $f$, may probably also be called an embedding.

The dual concept is that of effective epimorphism. See there for more discussion.

References

Exposition and examples:

• Qiaochu Yuan, Regular and effective monomorphisms and epimorphisms, 2012