# Definition:Morphism

## Definition

Let $\mathbf C$ be a metacategory.

A morphism of $\mathbf C$ is an object $f$, together with:

• A domain $\operatorname{dom} f$, which is an object of $\mathbf C$
• A codomain $\operatorname{cod} f$, also an object of $\mathbf C$

The collection of all morphisms of $\mathbf C$ is denoted $\mathbf C_1$.

If $A$ is the domain of $f$ and $B$ is its codomain, this is mostly represented by writing:

$f: A \to B$ or $A \stackrel{f}{\longrightarrow} B$

## Remark

A morphism is one of the two basic concepts of a metacategory, and therefore of category theory.

The other one is the notion of an object.

Thus in order to discuss a particular metacategory, it is necessary to specify what exactly its morphisms are.

Note that a morphism is defined to be an object. This should not be confused with an object of $\mathbf C$. Usually, the morphisms of a metacategory $\mathbf C$ are not also objects of $\mathbf C$.

## Also known as

Various other names for morphism include arrow and edge.

Awodey, in Category Theory, takes the arrow analogy further by proposing the term archery in place of category theory.