Definition:Epimorphism (Category Theory)

This page is about Epimorphism in the context of Category Theory. For other uses, see Epimorphism.

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.

Often, epimorphism is abbreviated to epi.

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

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.