# Definition:Epimorphism (Category Theory)

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*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**.

## Also known as

Often, **epimorphism** is abbreviated to **epi**.

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

## Also see

- Monomorphism, the dual notion

## 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**.

## Sources

- 2010: Steve Awodey:
*Category Theory*(2nd ed.) ... (previous) ... (next): $\S 2.1$: Definition $2.1$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**epimorhism**