Definition:Graph (Category Theory)
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Definition
A graph is an interpretation of a metagraph within set theory.
Let $\mathfrak U$ be a class of sets.
A metagraph $\GG$ is a graph if and only if:
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If the class $\mathfrak U$ is a set, then morphisms are functions, and the domain and codomain in the definition of a morphism are those familiar from set theory.
If $\mathfrak U$ is a proper class this is not the case, for example the morphisms of $\CC$ need not be functions.
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