Definition:Morphism

Definition
In the definition of a category, a morphism is just a name to put to a certain kind of object, as an abstraction of a mapping.

In some fixed category $\mathcal C$, a morphism of $\mathcal C$ has a specific definition, and may satisfy stronger properties.

In the latter case a morphisms can be thought of as homomorphisms that need not be mappings. One could equally take the view that a mapping need not be defined between sets (indeed in Gödel-Bernays set theory one must allow relations to operate between classes).

Abstract Category
A morphism or arrow is an object $f$, together with two objects $X = \operatorname{dom}f$ and $Y = \operatorname{cod}f$ called the domain and codomain of $f$ repectively, written $f : X \to Y$ or $X \stackrel{f}{\longrightarrow} Y$.

Note that a morphism is not necessarily a function, and $X$, $Y$ need not be sets.

Therefore, the terms on this page are axioms, and do not equate with the definitions from set theory with the same names.

Morphism of Graphs
Let $\mathcal G$, $\mathcal G'$ be graphs with vertices $V,V'$ and edges $E,E'$ respectively.

For an edge $a$ of a graph let $\partial_0$ and $\partial_1$ map to the source and destination of $a$ respectively.

A morphism of graphs $D : \mathcal G \to \mathcal G'$ associates:


 * To each vertex $v \in V$ a vertex $D_V(v) \in V'$
 * To each edge $e \in E$ and edge $D_E(e) \in E'$

such that for all edges $f \in \mathcal G$:


 * $D_V (\partial_0 f) = \partial_0 D_E (f)$ and $D_V (\partial_1 f) = \partial_1 D_E (f)$