# Definition:Monomorphism (Category Theory)

## 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.

## 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.