# 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.

That is, an epimorphism is a morphism which is right cancellable.

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.