# Definition:Monomorphism (Category Theory)

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

## Definition

Let $\mathbf C$ be a metacategory.

A **monomorphism** is a morphism $f \in \mathbf C_1$ such that:

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

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

That is, a **monomorphism** is a morphism which is left cancellable.

One writes $f: C \rightarrowtail D$ to denote that $f$ is a **monomorphism**.

## Also known as

Often, **monomorphism** is abbreviated to **mono**.

Alternatively, one can speak about a **monic** morphism to denote a **monomorphism**.

## Also see

- Epimorphism, the dual notion

## Linguistic Note

The word **monomorphism** comes from the Greek **morphe** (* μορφή*) meaning

**form**or

**structure**, with the prefix

**mono-**meaning

**single**.

Thus **monomorphism** means **single (similar) structure**.

## Sources

- 2010: Steve Awodey:
*Category Theory*(2nd ed.) ... (previous) ... (next): $\S 2.1$: Definition $2.1$