# Definition:Morphism

## Contents

## 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.

## Also see