Definition:Morphism
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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 Categories
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.
Concrete Categories
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 $D_V (\partial_1 f) = \partial_1 D_E (f)$
