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.
Also see
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): morphism