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How is it called a Rel-morphism $(f;A;B)$ such that: a. $f=\varnothing$; b. $f\ne\varnothing$?

Is there any special term for this?

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2 Answers 2

First things first: $\textbf{Rel}$ is an enriched category, and in more than one way.

  1. $\textbf{Rel}$ is a pointed category: that means it has a zero object and is enriched over the category of pointed sets. The empty relation $\emptyset : A \to B$ can be characterised as the unique relation $A \to B$ that factors through the zero object $\emptyset$, so we call it a zero morphism. Obviously, this property is preserved by composition on the left and on the right.

  2. $\textbf{Rel}$ is a locally posetal category: that means each hom-set is partially ordered, and composition is monotonic. The empty relation is clearly the bottom element of each hom-poset.

  3. Extending the above in a trivial way, $\textbf{Rel}$ is a 2-category: that means each hom-set is a category, and composition is functorial. The empty relation is clearly the initial object of each hom-category.

  4. $\textbf{Rel}$ is in fact enriched over itself, as discussed here, but to make this concrete, we should say that $\textbf{Rel}$ is enriched over the category of cocomplete join semilattices. (I say this instead of boolean algebra because composition only preserves joins.) As in (2), the empty relation is distinguished here as the bottom element of each hom-semilattice.

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It is called a zero morphism of Rel.

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