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 $A \stackrel f \longrightarrow B$

Motivation

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

A morphism is also known as:

an arrow

an edge.

Also see

• Definition:Object (Category Theory)
• Definition:Metacategory
• Definition:Metagraph
• Definition:Identity Morphism

• Results about morphisms can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): category: 2.
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): morphism
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): category: 2.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): morphism
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): morphism