Definition:Epimorphism (Category Theory)

Definition
Let $\mathbf C$ be a metacategory.

An epimorphism is a morphism $f \in \mathbf C_1$ such that:


 * $g \circ f = h \circ f \implies g = h$

for all morphisms $g, h \in \mathbf C_1$ for which these compositions are defined.

One writes $f: C \twoheadrightarrow D$ to denote that $f$ is an epimorphism.

Also known as
Often, epimorphism is abbreviated to epi.

Alternatively, one can speak about an epic morphism to denote an epimorphism.

Linguistic Note
The word epimorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix epi- meaning onto.

Thus epimorphism means onto (similar) structure.